Applications of the Integral

Topics

Problems 2

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Topics Problems 2

Problem : Apply the disk method to find the volume of the unit sphere obtained by revolving the region below the graph of f (x) = from x = - 1 to 1 about the x-axis.

Using the disk method, we have

Π()²dx	=	Π(1 - x²)dx
	=	Π -
	=

as expected.

Problem : Apply the shell method to find the volume of the solid obtained by revolving the region below the graph of f (x) = x² from x = 0 to 1 about the y-axis (this solid looks like a cylinder with a bowl carved out of the top).

The formula from the shell method gives

2Πx(x²)dx	=	2Πx³dx
	=	2Π\|₀¹
	=

Problem : Compute the volume of the square pyramid with base in the x = 0 plane with sides of length 10 and vertex at the point (5, 0), using the cross-sectional area method.

The cross-section of the pyramid in the plane perpendicular to the x-axis with a particular x-coordinate in the interval [0, 5] is a square with sides of length s(x) = 10 - 2x. Such a square has area equal to A(x) = s(x)² = (10 - 2x)² = 4x² - 40x + 100, so the volume of the pyramid is given by