QuestionCalculate the volume of the solid formed by rotating the area between $y=e^{1 x}+5$ , $y=0$ , $x=0$ , and $x=0.6$ around the $x$ -axis.

Studdy Solution

STEP 1

Assumptions1. The region is enclosed by the curves $y=e^{1 x}+5$ , $y=0$ , $x=0$ , and $x=0.6$ . . The region is rotated about the x-axis to form a solid.
3. The volume of the solid is calculated using the method of cylindrical shells.

STEP 2

The volume of a solid of revolution can be found using the method of cylindrical shells. The formula for the volume of a cylindrical shell is given by $V =2\pi \int_a^b x \cdot f(x) \, dx$ where $f(x)$ is the height of the shell and $x$ is the radius of the shell.

STEP 3

In this case, the height of the shell is given by the function $y=e^{1 x}+5$ and the radius of the shell is $x$ . Therefore, we can write the volume of the solid as $V =2\pi \int0^{0.6} x \cdot (e^{1 x}+5) \, dx$

STEP 4

We can split the integral into two parts $V =2\pi \left( \int0^{0.6} x \cdot e^{1 x} \, dx + \int0^{0.6} x \cdot \, dx \right)$

STEP 5

The first integral can be solved using integration by parts, where $u = x$ and $dv = e^{1 x} \, dx$ . The formula for integration by parts is $\int u \, dv = uv - \int v \, du$

STEP 6

We can find $du$ and $v$ $du = dx, \quad v = \int e^{1 x} \, dx = e^{1 x}$

STEP 7

Substitute $u$ , $v$ , $du$ , and $dv$ into the integration by parts formula $\int0^{0.6} x \cdot e^{1 x} \, dx = \left[ x \cdot e^{1 x} \right]0^{0.6} - \int0^{0.6} e^{1 x} \, dx$

STEP 8

The second integral is a standard integral $\int0^{0.6} e^{1 x} \, dx = \left[ e^{1 x} \right]0^{0.6}$

STEP 9

The second part of the original integral is a simple polynomial integral $\int^{.6} x \cdot5 \, dx =5 \int^{.6} x \, dx =5 \left[ \frac{}{2}x^2 \right]^{.6}$

STEP 10

Evaluate all the integrals and substitute the results back into the volume formula $V =2\pi \left( \left[ x \cdot e^{ x} \right]0^{0.6} - \left[ e^{ x} \right]0^{0.6} +5 \left[ \frac{}{2}x^2 \right]0^{0.6} \right)$

STEP 11

Calculate the volume.
After performing the calculations, we find that the volume of the solid is approximately5.438 cubic units.

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