Question

99 POINT QUESTION, PLUS BRAINLIEST!!!

Answer 1

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!

Answer 2

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