Use a deﬁnite integral to ﬁnd an expression that represents the area of the region between the given curve and the
x-axis on the interval [0, b].
-$$y = 12 \pi x ^ { 2 }$$
A) $- 4 \pi \mathrm { b } ^ { 3 }$
B) $- 24 \pi \mathrm { b }$
C) $4 \pi \mathrm { b } ^ { 3 }$
D) $24 \pi \mathrm { b }$

Question

Quizplus · Accepted Answer

To find the area between the curve and the x-axis on the interval [0, b], we need to integrate the equation of the curve with respect to x from 0 to b:

$$A = \int_0^b 12\pi x^2 dx$$

Then, we can use the power rule of integration to solve this:

$$A = 4\pi x^3 \Big|_0^b$$
$$A = 4\pi b^3 - 4\pi(0)^3$$
$$A = 4\pi b^3$$

Therefore, the expression that represents the area of the region between the curve and the x-axis on the interval [0, b] is 4πb3, which is answer choice C.

Use a Deﬁnite Integral to ﬁNd an Expression That Represents y=12πx2y = 12 \pi x ^ { 2 }y=12πx2

Use a Deﬁnite Integral to ﬁNd an Expression That Represents $y = 12 \pi x ^ { 2 }$