Find the area enclosed between f(x)=0.3x2+7 and g(x)=x from x=−4 to x=8.
Answers
f(x) is a parabola. It's lowest point occurs when x = 0. It is the point (0,7). At x = -4 and x=8 it has the values 11.8 and 26.2 respectively. That is, it contains the points (-4, 11.8) and (8,26.2).
From these we make a rough sketch (see attachment). This is a sketch and mine is very incorrect when it comes to scale but what matters here is which of the curves is on top, which is below and whether they intersect anywhere in the interval, so my rough sketch is good enough. From the sketch we see that f(x) is always above (greater than) g(x).
To find the area between the curves over the given interval we integrate their difference and since f(x) is strictly greater than g(x) we subtract as follows: f(x) - g(x). The limits of integration are the values -4 and 8 (the x-values between which we are looking for the area.
Now let's integrate:
[tex] \int\limits^{8}_ {-4}f(x)-g(x) \, dx = \int\limits^{8}_ {-4}.3 x^{2} +7-x \, dx [/tex]
The integral yields: [tex] [tex](\frac{.3 (8)^{3} }{3} +7(8)- \frac{ (8)^{2} }{2}) -(\frac{.3 (-4)^{3} }{3} +7(-4)- \frac{ (-4)^{2} }{2}) = 117.6[/tex] [/tex]
We evaluate this for 8 and for -4 subtracting the second FROM the first to get:
To find the area enclosed between two functions, we calculate the definite integral of the absolute difference between the functions over the given interval. The area is equal to the integral of ∣f(x) - g(x)∣, split into two parts where g(x) ≥ f(x) and g(x) < f(x), then added together. The enclosed area for the given functions is 82.26 square units.
Explanation:To find the area enclosed between the functions f(x) = 0.3x^2+7 and g(x) = x from x = -4 to x = 8, we need to calculate the definite integral of the absolute difference between the two functions over the given interval.
The area is equal to the integral of ∣f(x) - g(x)∣ from x = -4 to x = 8. We can split the integral into two parts where g(x) ≥ f(x) and g(x) < f(x), then evaluate each separately. Finally, we add the two areas together.
After evaluating the integrals, we find that the area enclosed between the two functions is 82.26 square units.
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Related Questions
Solve this problem using the proper formula
A gardener is installing a circular garden. The circumference of the circle 132 feet. What radius should he use to layout the circle if he uses p=22/7?
The radius is feet.
Answers
132 ft = 2(22/7)r
r = (132 ft)*7/44 = 21 ft
The radius is 21 feet.
Given that the garden should be in a circular shape.
And the circumference of the circle would be, C = 132 feet.
It says to find the radius of the circular garden, r = ?
We know the formula for Circumference of the circle is given as follows :-
Circumference of circle = 2π x radius
C = 2·π·r
Hint:- It says to use pi = 22/7.
[tex] 132=2*\frac{22}{7}*r \\\\
132=\frac{44}{7}*r \\\\
r=132*\frac{7}{44} \\\\
r= \frac{924}{44} \\\\
r= 21 \;feet [/tex]
Hence, Radius would be r = 21 feet.
which is smaller, half of 36 or one-seventh of 42
Answers
one seventh of 42 = 42/7 = 6
one seventh of 42 is smaller than half of 36 by 12
hope this helps
what is -10+2 because this is really confusing to me.
Answers
Hope this helps!
Let me know if i'm wrong.
what are the slope and why intercept shown on the graph below
Answers
Let's start by finding the slope of the line. You can calculate the slope by dividing the change in y-values by the change in x-values using the following formula:
[tex] \frac{ y_{1}- y_{2} }{ x_{1} - x_{2} } [/tex]
The plotted coordinates are (0, 5) and (4, -5); let's plug those into the formula:
[tex] \frac{5(-5)}{0-4} [/tex]
Simplify:
[tex] \frac{5+5}{-4} [/tex]
[tex] \frac{10}{-4} [/tex]
[tex] \frac{-5}{2} [/tex]
The slope is [tex] \frac{-5}{2} [/tex].
Now, the y-intercept is where the line intersects the y-axis, or when x equals 0. x equals 0 when y equals 5, so the y-intercept is 5.
I hope this helps!
y2 = -5
y1 = 5
x2 = 4
x1 = 0
Let's subtract. -5 - 5 is -10. 4 - 0 is 4. -10/4 is -2 1/2 in simplest form or -5/2 as a mixed number. We found the slope. The y-intercept is where the line crosses, where the x-axis is 0. In this case, we can see by the point given out that the slope intercept is 5. There. The slope is -5/2 and the slope-intercept is 5. The answer is B.
solve the inequality -×/2 <4
Answers
x > -8
_____
Multiplying or dividing by a negative number causes the direction of the relationship to change.
A rectangular prism with a volume of 6 cubic units is filled with cubes with side lengths of 1/2 units.
How many 1/2 unit cubes does it take to fill the prism?
Answers
48. If the side length is 1/2, cube it to find the volume of the cube. Then divide 6 by 1/8. Remember that if you are dividing a fraction, you multiply by the reciprocal, in this case 8. 6 times 8 is 48.
Answer:
48
Step-by-step explanation:
20 POINTS!!! Of 35 jewelers in her town, Naomi polled the 5 biggest to learn about the latest jewelery trends. Her sample is (biased; Not biased).The sample is (representative; not representative) PLZ ANSWER IN THAT ORDER!! I NEED THIS ASAP
Answers
Answer:
Biased / Not Representative
Step-by-step explanation:
The answer is biased because she is asking only the 5 biggest jewelers in her whole town, not the whole 35. It is also not representative because it is not representing allts of of the though every jeweler. Hope this helps <3
PS. Brainliest is wrong.
Answer:
I hope this helps
Step-by-step explanation:
The dimensions of an office conference room are 10 feet by 18 feet. If the conference room blueprint dimensions are 5 centimeters by 9 centimeters, what is the scale of the blueprint?
A. 1 centimeter = 9 feet
B. 1 centimeter = 2 feet
C. 1 centimeter = 5 feet
D. 1 centimeter = 10 feet
Answers
Jalen plots two integers on a horizontal number line. The leftmost integer is negative. Which must be true of the second integer?
The second integer is negative.
The second integer is positive.
The second integer is greater than the first integer.
The second integer is less than the first integer.
Answers
Answer:
c)The second integer is greater than the first integer
Step-by-step explanation:
In mathematics, a number line can be defined as a straight line with numbers placed at equal intervals or segments along its length.
A number line can be extended infinitely in any direction and is usually represented horizontally, with 0 in the middle and negative numbers on the left and positive numbers to the right of zero.
Thus a number line can be used to represent both positive and negative rational numbers as well as integers.
The numbers to the left are less than that to the right
Hence here the option c) is right
If a number is located farther right on the number line than another, then it is larger. True False
Answers
Answer:
it is TRUE my homei
Step-by-step explanation:
Which function is the inverse of f(x)=4x-1?
Answers
The function f⁻¹(x) = x/4 + 1/4 is the inverse of f(x)=4x-1
What is a function?The function is a type of relation, or rule, that maps one input to specific single output.
In mathematics, a function is an expression, rule, or law that describes the relationship between one variable (the independent variable) and another variable (the dependent variable). In maths and the physical sciences, functions are indispensable for formulating physical relationships.
The function is given as f(x)=4x-1
To determine the inverse of a function, switch the places of x and y (or f(x)).
f(x) = 4x - 1
y = 4x - 1
Switch the variables;
x = 4y - 1
Now, convert the equation back into y = mx + b form.
x + 1 = 4y
y = x/4 + 1/4
f⁻¹(x) = x/4 + 1/4
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plz help ill give you brainlist plz help
Answers
A = l * w
in this case A = 108 and w = 9
108 = 9l
l = 12 ft
the average adult human has approximately 2.5 x 10 ^ 13 red blood cells + 7 x 10 ^ 9 white blood cells about how many times as great is the number of red blood cells than the number of white blood cells
Answers
N = (red blood cells) / (white blood cells)
Substituting values we have:
N = (2.5 * 10 ^ 13) / (7 * 10 ^ 9)
N = 3571.428571
Nearest whole number:
N = 3571
Answer:
The number of red blood cells is 3571 as great as the number of white blood cells
Alex is decorating a large box with pieces of ribbon. He needs 7 sections of ribbon with a length of (3/4) yard. He is going to cut the 7 pieces from a roll of ribbon that is 30.2 yards long
Answers
SEE ATTACHED IMAGE.
For this case what you need to know is how much tape Alex needs.
We have then:
"He needs 7 sections of ribbon with a length of 3/4 yard"
In other words:
(7) * (3/4) = 5.25 yards.
Therefore, the amount of tape that will be left over will be:
30.2-5.25 = 24.95 yards.
Answer:
after Alex cuts the 7 pieces will remain 24.95 yards of ribbon
a puppy weighs 9/10 pound. it's mother weighs 8 times as much. how much does the mother weigh?
Answers
mrs. west is 14 years younger than her aunt. if mrs. west’s age in years is as much below 60 as her aunt’s age is over 40, how old is each?
Answers
Mrs. west is [tex]14[/tex] years younger than her aunt. if mrs. west’s age in years is as much below [tex]60[/tex] as her aunt’s age is over [tex]40[/tex], So, Mrs. West is [tex]43[/tex] years old, and her aunt is [tex]57[/tex] years old.
Let's represent Mrs. West's age as W and her aunt's age as A.
Given that Mrs. West is [tex]14[/tex] years younger than her aunt, we can write the equation:
[tex]W=A-14[/tex]
Also, it's given that Mrs. West's age is as much below [tex]60[/tex] as her aunt’s age is over [tex]40[/tex] . This can be represented as:
[tex]W=60-(A-40)[/tex]
Simplify the equation:
[tex]W=60-A+40\\W=100-A[/tex]
Now, we have two equations for [tex]W[/tex]:
[tex]W=A-14\\W=100-A[/tex]
Since both equations equal [tex]W[/tex], we can set them equal to each other:
[tex]A-14=100-A[/tex]
Now, solve for A:
[tex]A+A=100+14\\2A=114\\A=57[/tex]
Now, plug the value of [tex]A[/tex] into either equation to find [tex]W[/tex]:
[tex]W=57-14\\W=43[/tex]
So, Mrs. West is [tex]43[/tex] years old, and her aunt is [tex]57[/tex] years old.
COMPLETE QUESTION:
Mrs. west is [tex]14[/tex] years younger than her aunt. if mrs. west’s age in years is as much below [tex]60[/tex] as her aunt’s age is over [tex]40[/tex], how old is each?
What is greater than 1 kg
Answers
The center of a circle is at the origin. An endpoint of a diameter of the circle is at (-3, -4). How long is the diameter of the circle? 5, 10, or 25?
Answers
The diameter of the circle is 10.
What is the value of log0.5^16? a.–4.00 b.–0.25 c.1.51 d.2.41
Answers
Answer:
The correct option is a.
Step-by-step explanation:
The given expression is,
[tex]log(0.5)^{16}[/tex]
Use property of logarithm
[tex]loga^b=bloga[/tex]
[tex]log(0.5)^{16}=16\times log(0.5)[/tex]
The value of log(0.5) is -0.30103.
[tex]log(0.5)^{16}=16\times -0.30103[/tex]
[tex]log(0.5)^{16}=-4.81648[/tex]
The value of given expression is -4.81648.
If the final answer appromatted to preceding integer, then the correct option is -4.00.
What is the y-intercept of the line that has a slope of -1/2 and passes through the point (2, 3)?
Answers
y - 3 = -1/2(x) + 1
y = -1/2(x) + 4
y intercept is 4
answer
y intercept (0,4)
Indicate whether each of the following fractions is proper or improper. a. 4⁄16 b. 75⁄70 c. 2⁄15 d. 6⁄6
Answers
An improper fraction is a fraction where the numerator is larger than or equal to the denominator.
a. 4/16
The denominator is 16. The numerator is 4. The denominator is larger than the numerator; thus, this is a proper fraction.
b. 75/70
The denominator is 70. The numerator is 75. The numerator is larger than the denominator; thus, this is an improper fraction.
c. 2/15
The denominator is 15. The numerator is 2. The denominator is larger than the numerator; thus, this is a proper fraction.
d. 6/6
The denominator is 6. The numerator is 6. The numerator is equal to the denominator; thus, this is an improper fraction.
if 5 lemons cost £1 and a bag of 4 oranges cost £1.80, how much more does one orange cost than the lemon?
Answers
tickets to a play cost $11 each. There is also a service charge of $4 per order. Write ab equation for the relationship that gives the total cost y in dollars for an order of x tickets.
Answers
The equation representing the total cost of an order of tickets is y = 11x + 4, where y is the total cost and x is the number of tickets.
Explanation:The problem involves a constant service charge and a variable cost that depends on the number of tickets bought. In this case, the constant cost is the $4 service fee and the variable cost is the $11 per ticket. We express this relationship with the equation y = 11x + 4, where y is the total cost and x is the number of tickets purchased.
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tossing a number cube numbered from 1 to 6 and getting an odd number that is less than or equal to 3
Answers
2/6=1/3
At 7:45 a.m. Mike began walking to school. He arrived at his school at 8:20 a.m. How long did it take him to walk to school?
A. 15 minutes
B. 25 minutes
C. 35 minutes
D. 40 minutes
E. 65 minutes
Answers
7:45+15=8:00 = 15:00 minutes so far.
8:00+20= 8:20= 20+15= 35.
Your correct answer would be (option c).
35 is your answer!
Jim Abbott purchased a $60,000 recreational vehicle (RV) with a 40 percent markup on selling price. a. What was the amount of the dealer’s markup? Markup amount $ b. What was the dealer’s original cost? Original cost $
Answers
Hope this helped!
Answer:
24,000 dealer paid
mark up : 36,000
Step-by-step explanation:
60,000 * 0.40= 24,000
60,000-24,000= 36000
In certain county, the number of charter schools is 4vless than twice the numbervof alternative schools. We know that there are 48 charter schools in the county. How many alternative schools are in the county?
Answers
It is given that the number of charter schools is 4 less than twice the number of alternative schools. So we can write, the number of charter schools is 4 less than twice of x. In equation form this can be written as:
Number of Charter Schools = 2x - 4
It is given that number of charter schools is 48. So we can write:
48 = 2x - 4
2x = 52
x = 26
x was the number of alternative schools. Therefore, there are 26 alternative schools in the county.
- The number of charter schools is 4 less than twice the number of alternative schools.
- There are 48 charter schools in the country.
2. So, you have:
x=2y-4 (i)
x=48 (ii)
3. Then, when you substitute x=48 into the equation (i), you obtain:
x=2y-4
48=2y-4
4. Now, you must clear "y, as below:
48+4=2y
2y=52
y=52/2
y=26
How many alternative schools are in the county? The answer is: There are 26 alternative schools.
11x−6y=5 −4x−6y=13 How many solutions does this system have
Answers
2 times 2 plus 10 plus 200 plus 56
Answers
The order of operations requires the multiplication be done first.
= 4 + 10 + 200 + 56
The addition is done left to right.
= 14 + 200 + 56
= 214 + 56
= 270
Answer:
270
Step-by-step explanation:
2*2+10+200+56=270
solve for y
11x+7y=-35
Answers
11 x + 7 y = -35
Subtract 11 x from both sides:
7 y = -11 x - 35
Divide both sides by 7:
Answer: y = -(11 x)/7 - 5