Worksheet for Section 5.4

Secton 5.4 is about properties of definite integrals, summarized here in a series of theorems. The geometric interpretations of definite integrals suggest some specific properties of definite integrals, and many of the properties can be proven using the properties of limits and sums from earlier. Start with some properties about the limits on a definite integral:

• $$\int_a^a f(x)\, dx=0$$
• $$\int_a^b f(x)\, dx=-\int_b^a f(x)\, dx$$
• $$\int_a^b f(x)\, dx= \int_a^c f(x)\, dx+\int_c^b f(x)\, dx$$

For example, if $\int_0^2 f(x)\, dx=5$ and $\int_2^5 f(x)\, dx=8$, what is $\int_0^8 f(x)\, dx$?

The next few properties talk about constant multiples, sums, and differences of funcitons in definite integrals.

• $$\int_a^b k\, f(x)\, dx=k\int_a^b f(x)\, dx$$
• $$\int_a^b [f(x)\pm g(x)]\, dx= \int_a^b f(x)\, dx\pm\int_a^b g(x)\, dx$$

As an example, use these properties to determine $$\int_0^3 2+3x\, dx$$. Start by splitting up the integral using the properties above, then consider geometric interpretations of the regions in each part.

• if $f(x)$ is an even function, then $$\int_{-a}^a f(x)\, dx = 2\int_0^a f(x)\, dx$$
• if $f(x)$ is an odd function, then $$\int_{-a}^a f(x)\, dx = 0$$

So, for example, since $f(x)=\sin x$ is an odd function, $$\int_{-\pi}^pi \sin x\, dx=0$$ -- sketch a graph of $f(x)=\sin x$ on the interval $[-\pi,\pi]$ to see why. On the other hand, $f(x)=4-x^2$ is an even function, so $$\int_{-2}^2 4-x^2\, dx=2\int_0^2 4-x^2\, dx$$ -- again, sketch a graph of $f(x)=4-x^2$ to see why. What can you say about $$\int_{-\pi}^pi \vert\sin x\vert\, dx$$?

As an example combining some of these properties, suppose you know that $$\int_0^1 \cos(x^2)\, dx=0.90$$ and $$\int_0^{1.25}\cos(x^2)\, dx=0.98$$. What is the value of $$\int_{-1.25}^{-1} 2\cos(x^2)\, dx$$?

The next few properties look at comparing the values of definite integrals if you know something about comparisons of the functions.

• if $f(x)\geq 0$ on $[a,b]$ then $$\int_a^b f(x)\, dx\geq 0$$
• if $m\leq f(x)\leq M$ on $[a,b]$, then $$m(b-a)\leq \int_a^b f(x)\, dx\leq M(b-a)$$
• if $f(x)\geq g(x)$ on $[a,b]$ then $$\int_a^b f(x)\, dx\geq \int_a^b g(x)\, dx$$

For example, sketch the graph of $f(x)=\sin(x^2)$ and explain why $$\int_0^{\sqrt{\pi}}\sin(x^2)\, dx\leq\sqrt{\pi}$$.

Recall from the example in the last section that $$\int_0^2 x^2\, dx=\frac53$$. Using this, along with some geometry as above and some of these basic properties, compute
$$\int_0^2 -x^2+2x+2\, dx$$.

You can also use definite integrals to find the area of a region between two curves. Suppose the two curves between which you want to find the area are given by $y=f(x)$ and $y=g(x)$, where the graph of $f(x)$ lies above the graph of $g(x)$. Then the area between the curves is given by the following formula:

$$\int_a^b f(x)-g(x)\, dx$$

The limits $a$ and $b$ for this integral are often determined by where the curves intersect. For each of the pairs of functions given below, find the area between the curves:

$$f(x) = 2-x^2 f(x) = x^3$$

$$g(x) = x g(x) = 4x$$

For each of these, you will need to start by finding out where the curves intersect. In the second example, be careful -- there are actually two regions for which you must find areas.