january 06, 2016 let's warm up with a warmup!!€¦ · january 06, 2016 let's warm up...

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)

January 06, 2016

January 06, 2016

January 06, 2016

January 06, 2016

January 06, 2016

January 06, 2016

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