(a) Estimate the volume of the solid that lies below the...

(a) Estimate the volume of the solid that lies below the surface $ z = xy $ and above the rectangle $$ R = \{(x, y) \mid 0 \le x \le 6, 0 \le y \le 4 \} $$ Use a Riemann sum with $ m = 3 $, $ n = 2 $, and take the sample point to be the upper right corner of each square. (b) Use the Midpoint Rule to estimate the volume of the solid in part (a).

(a) Estimate the volume of the solid that lies below the surface $ z = xy $ and above the rectangle
$$ R = \{(x, y) \mid 0 \le x \le 6, 0 \le y \le 4 \} $$
Use a Riemann sum with $ m = 3 $, $ n = 2 $, and take the sample point to be the upper right corner of each square.
(b) Use the Midpoint Rule to estimate the volume of the solid in part (a).

Transcript

00:01 For this problem, we are going to use rhyme and sums to approximate the volume of a solid.

00:07 It's going to lie below the surface of z equals x, y, and above the square r.

00:14 Now, remember, for these rhyme and sums, we're going to, we're approximating three -dimensional blocks that are filling up this volume as best that we can.

00:24 So let's take a look at the base at r.

00:27 So i'm just going to sketch this here.

00:30 So here's r.

00:31 And i'm going over for x equals 6 and y equals 4.

00:38 Now, how many squares are we breaking r into? because it's a nice little rectangular area here.

00:46 Depending on how many squares we break this up into, it'll change the way we've come up with our values.

00:53 So we're told that we want m to equal 3 and n to equal 2.

00:57 So m, that means i'm going to break x up into three squares.

01:02 So two and four are going to be where i break this up.

01:08 N is two, which means i'm breaking my y in half.

01:11 So i'm going to do that here from two and four.

01:14 So that gives me six total blocks.