MA005: Calculus I

If one of the endpoints $a$ or $b$ of the interval $[a, b]$ changes, then the value of the integral $\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}(t) \mathrm{dt}$ typically changes. A definite integral of the form $\int_{\mathrm{a}}^{x} \mathrm{f}(t) \mathrm{dt}$ defines a function of $x$ , and functions defined by definite integrals in this way have interesting and useful properties. The next examples illustrate one of them: the derivative of a function defined by an integral is closely related to the integrand, the function "inside" the integral.

Example 2: For the function $f(t)=2$, define $\mathrm{A}(x)$ to be the area of the region bounded by $f$, the t–axis, and vertical lines at $t = 1$ and $t = x$ (Fig. 7).

(a) Evaluate $A(1)$, $A(2)$, $A(3)$, $A(4)$.

(b) Find an algebraic formula for $\mathrm{A}(x)$ for $x \geq 1$.

(c) Calculate $\frac{\mathrm{d}}{\mathrm{dx}} \mathrm{A}(x)$.

(d) Describe $\mathrm{A}(x)$ as a definite integral.

Solution :

(a) $\mathrm{A}(1)=0, \mathrm{~A}(2)=2, \mathrm{~A}(3)=4, \mathrm{~A}(4)=6$.

(b) $\mathrm{A}(\mathrm{x})=\text { area of a rectangle }=(\text { base }) \cdot(\text { height })=(x-1) \cdot(2)=2 x-2$ .

(c) $\frac{\mathrm{d}}{\mathrm{dx}} \mathrm{A}(x)=\frac{\mathrm{d}}{\mathrm{dx}}(2 x-2)=2$.

(d) $\mathrm{A}(\mathrm{x})=\int_{1}^{x} 2 \mathrm{dt}$.

Practice 3: Answer the questions in the previous Example for $\mathrm{f}(x)=3$.

Example 3: For the function $\mathrm{f}(t)=1+t$, define $\mathrm{B}(x)$ to be the area of the region bounded by the graph of $f$, the t–axis, and vertical lines at $t = 0$ and $t = x$ (Fig. 8).

(a) Evaluate $B(0)$, $B(1)$, $B(2)$, $B(3)$.

(b) Find an algebraic formula for $\mathrm{B}(x)$ for $x \geq 0$.

(c) Calculate $\frac{\mathrm{d}}{\mathrm{dx}} \mathrm{B}(x)$.

(d) Describe $\mathrm{B}(x)$ as a definite integral.

Solution:

(a) $\mathrm{B}(0)=0, \mathrm{~B}(1)=1.5, \mathrm{~B}(2)=4, \mathrm{~B}(3)=7.5$.

(b) $\mathrm{B}(\mathrm{x})=\operatorname{area} \text { of trapezoid }=(\text { base }) \cdot(\text { average height })=(\mathrm{x}) \cdot\left(\frac{1+(1+x)}{2}\right)=\mathrm{x}+\frac{x^{2}}{2}$.

(c) $\frac{\mathrm{d}}{\mathrm{dx}} \mathrm{B}(x)=\frac{\mathrm{d}}{\mathrm{dx}}\left(x+\frac{x^{2}}{2}\right)=1+x$.

(d) $\mathrm{B}(\mathrm{x})=\int_{0}^{x} 1+t \mathrm{dt}$.

Practice 4: Answer the questions in the previous Example for $\mathrm{f}(t)=2 t$.

A curious "coincidence" appeared in each of these Examples and Practice problems: the derivative of the function defined by the integral was the same as the integrand, the function "inside" the integral. Stated another way, the function defined by the integral was an "antiderivative" of the function "inside" the integral. In section 4.4 we will see that this "coincidence" is a property of functions defined by the integral. And it is such an important property that it is called The Fundamental Theorem of Calculus, part I. Before we go on to the Fundamental Theorem of Calculus, however, there is an "existence" question to consider: Which functions can be integrated?