Approximating the volume under a surface

Let $z = f(x,y)$ be a function of two variables that is continuous on a rectangular region $R \subseteq \mathbb{R}^2$,

$$ R = \Big\{ (x,y) \mid a \leq x \leq b, c \leq y \leq d \Big\}. $$
This rectangle can be referred to as

$$ R = [a,b] \times [c,d]. $$
The convention is that the intervals are in alphabetical order: $x$ then $y$.

We begin by considering a geometric argument for approximating the volume of the solid region under the portion of the graph of $f$ over $R$. Moving forward, we'll refer to the portion of this surface as $\Gamma(f,R)$ and the solid region as $E$.

The exact volume under the surface is obtained by taking some limits and obtaining a Riemann sum.

Video Lecture

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1. Example

Use the Riemann sum definition to compute

$$ \int\!\!\!\int_R 3x^2 - 2y \, dA, $$
where $R = [0,1] \times [0,2]$.

Check your answer

$$-2$$

Video solution

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Iterated Integrals

In most cases, we will not need to compute these integrals by using Riemann sum definition. This is of course a consequence of the Fundamental Theorem of Calculus. In the following video, pay special attention to Fubini's Theorem. You will need to know this theorem and its limitations.

Video Lecture

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2. Example

Compute the iterated integral.

$$ \int_0^1 \int_0^1 \big(x + y\big)^2 \, dx\, dy. $$

Check your answer

$$\frac{7}{6}$$

Video solution

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3. Example

Compute the integral two ways, and verify that the conclusion of Fubini's Theorem holds.

$$ \int\!\!\!\int_R y \sin(xy) \,dA, $$
where $R = [1,2] \times [0,\pi]$.

Check your answer

Regardless of the order that you integrate, you should get $0$. But, one way is much easier than the other!

Video solution

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