January 06, 2016

Let's warm up with a warmup!!1) Use the trapezoid rule with n = 4 to approximate the area under the curve y = 6 - x2 for 0 ≤ x ≤ 2

2) Integrate

3) Solve dy/dx = x2y for y, given y(0) = 3 and y > 0.

4) The region bounded by y = x, y = 0 and x = 4 is revolved about the line x = 6. Find the volume of the solid.

1) A = (1/2)(1/2)[(6 + 23/4) + (23/4 + 5) + (5 + 15/4) + (15/4 + 2)] = 37/4 or 9.25

2)

3)

4)

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7.2b Volumes by Cross Sections!!

At the end of this lesson you will be able to:

• Calculate the volume of a solid using the cross section of the solid

January 06, 2016

What on earth do these shapes look like?

Let's see!

Volume by Cross Section!

cross sections are perpendicular to x-axis => dx

cross sections are perpendicular to y-axis => dy

January 06, 2016

ex) The base of a solid is the region bounded by y = 1+ lnx and y = x - 1. Find the volume of the solid if the cross sections are:

a) squares perpendicular to the x-axis

b) equilateral triangles parallel to the y-axis

c) rectangles where each height is twice the width, perpendicular to the y-axis

January 06, 2016

You try! The base of a solid is bounded by y = x2 and y = 4. Find the volume of the solid if the cross sections are:

1) squares parallel to the x-axis

2) isosceles right triangles with the hypotenuse on the base perpendicular to the y-axis

3) semicircles parallel to the y-axis

4) isosceles right triangles with one leg on the base perpendicular to the x-axis

January 06, 2016

What have we learned?

• Can I find the volume of a solid with consistent geometrical cross sections?

January 06, 2016