November 19, 2019

Warmup (review)1) Find the surface area of the portion of z = 2 x2 + y2 inside the cylinder x2 + y2 = 4 (no calculators!)

2) Find the center of mass of the lamina bounded by y = 6 - x/2 - x2/4 in the first quadrant if ρ = k (calculators permitted)

3) Find the volume of the solid bounded on top by z = sin(y) + 3 and on the bottom by the triangle in the xy-plane with corners at (-1, 1), (0, -2), and (3, 4) (calculators permitted)

November 19, 2019

15.6 Triple Integrals!!

Learning Target• I can use a triple integral to find the

volume of a solid• I can find the center of mass and

moments of inertia of a solid

November 19, 2019

The triple integral of f over Q is defined as:

where ||Δ|| is the length of the longest diagonal of the boxes in the partition and ΔVi is the volume of the ith box.

So to recap, with one integral we are taking the limit as one dimension approaches 0 (such as the width of a rectangle, think of rectangles that become lines) so if the integrand is the height, this gives us exact area.

With two integrals, we are taking the limit as two dimensions approach 0 (think of squares that become dots) so with no integrand, this also gives us exact area. With an integrand of the height, this gives us volume.

With three integrals, we are taking the limit as three dimensions approach 0 (think of cubes that become dots) so with no integrand, this gives us exact volume. With an integrand, we can use this to study concepts such as work and force that use volume (along with other factors like density).

November 19, 2019

Let's just make sure we can evaluate a triple integral.

no calculators!

November 19, 2019

Use triple integrals to find the volume of the ellipsoid given by 4x2 + 4y2 + z2 = 16.

Warning: your calculator might want to include a sin(∞) in the answer. Just treat this as 1. 🤔

November 19, 2019

How Dylan Yang feels about triple integrals

How you will feel about triple integrals by the end of this unit

November 19, 2019

Let's review! (All integrals may be done on a calculator.)Suppose f(x) = x3/3 and g(x) = 4x - x2

Consider the closed region in the first quadrant bounded by f(x) and g(x):1) Using a single integral, find the area of the region2) Using a double integral, find the area of the region3) Find the volumes of the solids created if the region is rotated

a) about the x-axisb) about x = 4

4) Find the perimeter of the region5) Find the centroid if the region is a planar lamina and ρ is constant6) Find the centroid if the region is a planar lamina and ρ = x + 3y

November 19, 2019

What have we learned?• Can I evaluate a triple integral and use it to find

the volume of a solid?