Math / Calculus

QuestionSet up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified li $y=\sin (x), \quad y=0, \quad 0 \leq x \leq \pi ; \quad$ about $y=-4$ $\square$ $\int d x$ Need Help? Read It

Studdy Solution

STEP 1

1. We are finding the volume of a solid of revolution.
2. The region is bounded by $y = \sin(x)$ , $y = 0$ , and $0 \leq x \leq \pi$ .
3. The axis of rotation is $y = -4$ .

STEP 2

1. Understand the geometry of the problem.
2. Identify the method for setting up the integral.
3. Set up the integral expression for the volume.

STEP 3

The region bounded by $y = \sin(x)$ , $y = 0$ , and $0 \leq x \leq \pi$ is a segment of the sine wave from $x = 0$ to $x = \pi$ .

STEP 4

The problem involves rotating the region around the line $y = -4$ , which is below the x-axis. We will use the method of cylindrical shells or the washer method.

STEP 5

Since the axis of rotation is horizontal and not on the boundary, we use the washer method. The outer radius is the distance from $y = -4$ to $y = 0$ , which is $4$ . The inner radius is the distance from $y = -4$ to $y = \sin(x)$ , which is $4 + \sin(x)$ .

STEP 6

The volume $V$ of the solid is given by the integral:
$V = \pi \int_{0}^{\pi} \left[ (4 + \sin(x))^2 - (4)^2 \right] \, dx$
The integral expression for the volume is:
$\pi \int_{0}^{\pi} \left[ (4 + \sin(x))^2 - 16 \right] \, dx$

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QuestionSet up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified li $y=\sin (x), \quad y=0, \quad 0 \leq x \leq \pi ; \quad$ about $y=-4$ $\square$ $\int d x$ Need Help? Read It

Studdy Solution

STEP 1

1. We are finding the volume of a solid of revolution.
2. The region is bounded by $y = \sin(x)$ , $y = 0$ , and $0 \leq x \leq \pi$ .
3. The axis of rotation is $y = -4$ .

STEP 2

1. Understand the geometry of the problem.
2. Identify the method for setting up the integral.
3. Set up the integral expression for the volume.

STEP 3

The region bounded by $y = \sin(x)$ , $y = 0$ , and $0 \leq x \leq \pi$ is a segment of the sine wave from $x = 0$ to $x = \pi$ .

STEP 4

The problem involves rotating the region around the line $y = -4$ , which is below the x-axis. We will use the method of cylindrical shells or the washer method.

STEP 5

Since the axis of rotation is horizontal and not on the boundary, we use the washer method. The outer radius is the distance from $y = -4$ to $y = 0$ , which is $4$ . The inner radius is the distance from $y = -4$ to $y = \sin(x)$ , which is $4 + \sin(x)$ .

STEP 6

The volume $V$ of the solid is given by the integral:
$V = \pi \int_{0}^{\pi} \left[ (4 + \sin(x))^2 - (4)^2 \right] \, dx$
The integral expression for the volume is:
$\pi \int_{0}^{\pi} \left[ (4 + \sin(x))^2 - 16 \right] \, dx$

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Studdy Solution

STEP 1

1. We are finding the volume of a solid of revolution.2. The region is bounded by y=sin⁡(x) y = \sin(x) y=sin(x), y=0 y = 0 y=0, and 0≤x≤π 0 \leq x \leq \pi 0≤x≤π.3. The axis of rotation is y=−4 y = -4 y=−4.

STEP 2

1. Understand the geometry of the problem.2. Identify the method for setting up the integral.3. Set up the integral expression for the volume.

STEP 3

The region bounded by y=sin⁡(x) y = \sin(x) y=sin(x), y=0 y = 0 y=0, and 0≤x≤π 0 \leq x \leq \pi 0≤x≤π is a segment of the sine wave from x=0 x = 0 x=0 to x=π x = \pi x=π.

STEP 4

The problem involves rotating the region around the line y=−4 y = -4 y=−4, which is below the x-axis. We will use the method of cylindrical shells or the washer method.

STEP 5

STEP 6

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1. We are finding the volume of a solid of revolution.
2. The region is bounded by $y = \sin(x)$ , $y = 0$ , and $0 \leq x \leq \pi$ .
3. The axis of rotation is $y = -4$ .

1. Understand the geometry of the problem.
2. Identify the method for setting up the integral.
3. Set up the integral expression for the volume.

The region bounded by $y = \sin(x)$ , $y = 0$ , and $0 \leq x \leq \pi$ is a segment of the sine wave from $x = 0$ to $x = \pi$ .

The problem involves rotating the region around the line $y = -4$ , which is below the x-axis. We will use the method of cylindrical shells or the washer method.