Question

A region is bounded by y=e−3x, the x-axis, the y-axis and the line x = 3. If the region is the base of a solid such that each cross section perpendicular to the x-axis is an equilateral triangle, set up the integral that would find the volume of that solid.

Answer 1

Answer:

0.333$\pi$ units³

Step-by-step explanation:

Think process:

The equation is given as y = $e^{-3x}$

Let, y = f (x)

Therefore, $f (x) = e^{-3x}$

We know that the limits are y-axis and x= 3

Y-axis: x= 0

then limits are given as x= 0 and x = 3

Integrating  gives:

$\int\limits^3_0 {e^{-3x} } \, dx$ = $\frac{-1}{3}e^{-3x}$ + C

calculating from x= 0 to x = 3, we know volume is given by $\pi \int\limits^a_b {f(x)} \, dx$

= $\pi$[ $\frac{-1}{3} e^{-9} - (\frac{-1}{3} e^{0})$]

= $\pi$[0.000041136 + 1/3]

= 0.333$\pi$ units³