Volume of a Solid with a Definite Integral

The video tutorial explains how to calculate the volume of a solid with a base defined by region R, enclosed by the curve y = 4√(9-x) and the axes in the first quadrant. The solid's cross-sections, perpendicular to the y-axis, are rectangles with a base in region R and height y. The tutorial guides viewers through visualizing the solid, setting up the integral, and solving for x in terms of y to express the volume as a definite integral from y = 0 to y = 12.

What is the region R defined by in the problem?

• y = 4 * sqrt(9 - x) and the axes in the first quadrant

• y = 3 * sqrt(9 - x) and the axes in the fourth quadrant

• y = 3 * sqrt(9 - x) and the axes in the second quadrant

• y = 4 * sqrt(9 - x) and the axes in the third quadrant

What shape are the cross-sections of the solid?

• Triangles

• Rectangles

• Circles

• Squares

What is the base of the rectangle in the cross-section?

• The x value corresponding to y

• The y-axis

• The x-axis

• The y value corresponding to x

What is the height of the rectangle in the cross-section?

• The x value

• The y value

• The z value

• The t value

How is the depth of an infinitesimal slice represented?

• dt

• dz

• dy

• dx

What is the expression for the volume of a slice?

• x * y * dx

• z * y * dz

• t * x * dt

• y * x * dy

What is the expression for x in terms of y?

• x = 16 - y^2/9

• x = 9 - y^2/16

• x = y^2/16 - 9

• x = 9 - 16/y^2

What is the integral's lower limit for y?

• y = 0

• y = 1

• y = 16

• y = 9

What is the integral's upper limit for y?

• y = 16

• y = 9

• y = 4

• y = 12

What is the final step to find the total volume of the solid?

• Multiply the expression by a constant

• Integrate the expression from y = 0 to y = 12

• Divide the expression by a constant

• Differentiate the expression