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Revolve the region bounded by
y=e^(x), the
x-axis,
x=0, and
x=1 about the line
y=4. Find the volume.

V=[?]

Revolve the region bounded by $y=e^{x}$ , the $x$ -axis, $x=0$ , and $x=1$ about the line $y=4$ . Find the volume. $\newline$ $V=[?]$

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Find equations of tangent lines using limits

Full solution

Q. Revolve the region bounded by

y=e^{x}

, the

x

-axis,

x=0

, and

x=1

about the line

y=4

. Find the volume.

\newline

V=[?]

Set up integral for volume: First, we need to set up the integral for the volume using the disk method since we're revolving around a horizontal line. The volume of a solid of revolution is given by the integral of $\pi(\text{outer radius}^2 - \text{inner radius}^2)$ from $a$ to $b$ .
Determine outer and inner radius: The outer radius is the distance from $y=4$ to $y=e^{x}$ , which is $4 - e^{x}$ . The inner radius is $0$ because the region is bounded by the x-axis ( $y=0$ ). So, we only need to consider the outer radius.
Calculate volume integral: Now we set up the integral: $V = \pi \int_{0}^{1} (4 - e^{x})^2 \, dx$ . This will give us the volume of the solid.
Expand and simplify expression: We need to expand $(4 - e^{x})^2$ to $(16 - 8e^{x} + e^{2x})$ before integrating.
Integrate term by term: Now we integrate: $V = \pi \cdot [\int_{0}^{1} 16 \, dx - \int_{0}^{1} 8e^{x} \, dx + \int_{0}^{1} e^{2x} \, dx]$ .
Evaluate limits of integration: Integrating term by term, we get $V = \pi \cdot [16x - 8e^{x} + (\frac{1}{2})e^{2x}]$ evaluated from $0$ to $1$ .
Simplify the expression: Plugging in the limits of integration, we get $V = \pi \cdot [(16\cdot 1 - 8e^{1} + (1/2)e^{2\cdot 1}) - (16\cdot 0 - 8e^{0} + (1/2)e^{2\cdot 0})]$ .
Combine like terms: Simplify the expression: $V = \pi \cdot [16 - 8e + (\frac{1}{2})e^2 - (0 - 8\cdot 1 + (\frac{1}{2})\cdot 1)]$ .
Calculate numerical value: Further simplification gives us $V = \pi \cdot [16 - 8e + (\frac{1}{2})e^2 - 8 + (\frac{1}{2})]$ .
Calculate numerical value: Further simplification gives us $V = \pi \times [16 - 8e + (\frac{1}{2})e^2 - 8 + (\frac{1}{2})]$ .Combine like terms: $V = \pi \times [8 - 8e + (\frac{1}{2})e^2 + (\frac{1}{2})]$ .
Calculate numerical value: Further simplification gives us $V = \pi \times [16 - 8e + (1/2)e^2 - 8 + (1/2)]$ .Combine like terms: $V = \pi \times [8 - 8e + (1/2)e^2 + (1/2)]$ .Now we calculate the numerical value: $V = \pi \times [8 - 8\times2.71828 + (1/2)\times2.71828^2 + (1/2)]$ .

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