Question

99 POINT QUESTION, PLUS BRAINLIEST!!!
(Please answer genuinely, and do not answer just for points, if you do, your answer will be deleted, and those points you earned will be taken away...)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
THIS IS CALCULUS NOT BASIC MATH...

7.) Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the y-axis.
y = x^7, x = 0, y = 128
A • V=pi*(integral of 0-128) y^(2/7) dy = (3584/9)pi
B • V=pi*(integral of 0-128) y^(2/7) dy = (1792/9)pi
C • V=pi*(integral of 0-7) y^(1/7) dy = (1792/9)pi
D • V=pi*(integral of 0-7) y^(1/7) dy = (3584/9)pi

** Please show all of you work, if you do not, your answer will be deleted, and the points you earned will be taken away...

Answer 1

Each answer choice has dy in it, indicating that we'll need to convert the function of x into a function of y. To do this, solve for x. Raise both sides to the 1/7th power to cancel the exponent. This is the same as taking the 7th root of both sides

y = x^7
(y)^(1/7) = (x^7)^(1/7)
y^(1/7) = x
x = y^(1/7)

Imagine that point P is any point on the y axis that is between (0,0) and (0,128), which is the interval we wish to integrate along. The horizontal distance from P to the curve x = y^(1/7) is exactly equal to y^(1/7). If we rotate the curve around the y axis, then we will form disks of radius y^(1/7).

For any given y value in the interval [0,128], we'll have circles with area pi*r^2 = pi*(y^(1/7))^2 = pi*y^(2/7)

So our answer is either A or B based on the result we just got.

Let's integrate to find out what the volume would be. Use the power rule for integration.

$V = \pi\int_{0}^{128}y^{2/7}dy$

$V = \pi*\left[\frac{1}{2/7+1}y^{2/7+1}+C\right]_{0}^{128}$

$V = \pi*\left[\frac{1}{2/7+7/7}y^{2/7+7/7}+C\right]_{0}^{128}$

$V = \pi*\left[\frac{1}{9/7}y^{9/7}+C\right]_{0}^{128}$

$V = \pi*\left[\frac{7}{9}y^{9/7}+C\right]_{0}^{128}$

$V = \pi*\left[\left(\frac{7}{9}(128)^{9/7}+C\right)-\left(\frac{7}{9}(0)^{9/7}+C\right)\right]$

$V = \frac{3584}{9}\pi$

So the final answer is choice A

Answer 2

Hello!

I believe the answer is A) V=pi*(integral of 0-128) y^(2/7) dy = (3584/9)pi

I have the work on a piece of paper but I don't know how to post it with this :/

I hope it helps!