Calculus AB/BC 8.7 Volumes with Cross Sections: Squares and Rect

A region is bounded by y = x^2 and y = sqrt(x). If each cross-section perpendicular to the x-axis is a square, which integral represents the volume of the solid?

• Integral from 0 to 1 of (x^2 - sqrt(x))^2 dx

• Integral from 0 to 1 of (sqrt(x) - x^2)^2 dx

• Integral from 0 to 1 of (sqrt(x) - x^2) dx

• Integral from 0 to 1 of (x^2 + sqrt(x))^2 dx

How is the volume of a solid with known cross-sections calculated?

• By integrating the perimeter of the cross-section.

• By integrating the area of the cross-section.

• By multiplying the area of the base by the height.

• By finding the derivative of the cross-section's area.

In the formula for the volume of a solid with square cross-sections, V = ∫[a,b] s^2 dx, what does 's' represent?

• The area of the square cross-section.

• The height of the solid.

• The side length of the square cross-section.

• The perimeter of the square cross-section.

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What is the correct integral setup to find the volume of a solid whose base is bounded by y=sqrt(x) and y=x^2, and whose square cross-sections are taken perpendicular to the y-axis?

• ∫[0 to 1] (sqrt(y) - y^2) dy

• ∫[0 to 1] (sqrt(y) - y^2)^2 dy

• ∫[0 to 1] (y^2 - sqrt(y))^2 dy

• ∫[0 to 1] (sqrt(x) - x^2)^2 dx

A solid's base is bounded by y=x^3, y=0, and x=2. If the cross-sections are rectangles perpendicular to the x-axis, and the height of each rectangle is 2 times its width, which integral represents the volume of the solid?

• ∫[0 to 2] (x^3) dx

• ∫[0 to 2] (x^3)^2 dx

• ∫[0 to 2] (x^3)(2x^3) dx

• ∫[0 to 2] (2x^3)^2 dx

For the same solid base bounded by y=x^3, y=0, and x=2, if the cross-sections are rectangles perpendicular to the y-axis with a constant height of 6, what is the correct integral for its volume?

• ∫[0 to 8] (2 - cube_root(y)) * 6 dy

• ∫[0 to 2] (2 - cube_root(y)) * 6 dy

• ∫[0 to 8] (2 - y^3) * 6 dy

• ∫[0 to 8] (2 - cube_root(y))^2 * 6 dy

The graphs of y = x^2 - 4 and y = 2x - x^2 create a bounded area that is the base of a solid. If this solid has square cross-sections perpendicular to the x-axis, what is the integral setup to find its volume?

• ∫[-1 to 2] ((x^2 - 4) - (2x - x^2))^2 dx

• ∫[-1 to 2] ((2x - x^2) - (x^2 - 4))^2 dx

• ∫[-2 to 2] ((2x - x^2) - (x^2 - 4))^2 dx

• ∫[-1 to 2] (2x - x^2 - x^2 - 4)^2 dx