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