9. Graphical Applications of Integrals

Introduction to Volume & Disk Method

9. Graphical Applications of Integrals

Introduction to Volume & Disk Method

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Multiple Choice

Find the volume of the solid obtained by rotating the region bounded by $y=x+4$ , $y=0$ , $x=1$ & $x=5$ .

$\frac{854}{3}\pi$

$4\pi$

$\frac{604}{3}\pi$

$604\pi$

Verified step by step guidance

Identify the region to be rotated: The region is bounded by the lines y = x + 4, y = 0, x = 1, and x = 5.

Determine the axis of rotation: The problem implies rotation around the x-axis.

Set up the integral for the volume using the disk method: The formula for the volume of a solid of revolution is $ V = \int_{a}^{b} \pi [R(x)]^2 \, dx $, where $ R(x) $ is the radius of the disk at a given x.

Find the radius function $ R(x) $: Since the region is rotated around the x-axis, $ R(x) = y = x + 4 $.

Evaluate the integral: Substitute $ R(x) = x + 4 $ into the volume formula and integrate from x = 1 to x = 5: $ V = \int_{1}^{5} \pi (x + 4)^2 \, dx $.

5:38m

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Introduction to Volume & Disk Method practice set

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Struggling with Business Calculus?

Find the volume of the solid obtained by rotating the region bounded by y=x+4y=x+4y=x+4, y=0y=0y=0, x=1x=1x=1 & x=5x=5x=5.

Watch next

Introduction to Volume & Disk Method practice set

Find the volume of the solid obtained by rotating the region bounded by $y=x+4$ , $y=0$ , $x=1$ & $x=5$ .