Chapter 6.2, Problem 40E

(a)

To determine

To Find: Points of discontinuity of the integrand on the interval of integration.

Answer to Problem 40E

Function is discontinuous at $x=3$

Explanation of Solution

Given information:

  ${{\displaystyle \int }}_{-5}^6\frac{9-x^2}{x-3}dx$

Calculation:

Function is not defined at $x=3$ because at this value of x, function is of indeterminate form.

Hence, function is discontinuous at

  $x=3$

(b)

To determine

To Evaluate: The given integral.

Answer to Problem 40E

The integral of the given is $\frac{77}{2}$ .

Explanation of Solution

Given information:

  ${{\displaystyle \int }}_{-5}^6\frac{9-x^2}{x-3}dx$

Calculation:

For any integral function,

  ${{\displaystyle \int }}_a^{b}f(x)dx=\left(areaoftriangleabovethex-axis\right)–\left(areaoftrianglebelowthex-axis\right)$

The graph of the function:

  

So,

  ${{\displaystyle \int }}_{-5}^6\frac{9-x^2}{x-3}dx=\left(areaoftriangleabovethex-axis\right)–\left(areaoftrainglebelowthex-axis\right)$

  $=\frac{1}{2}\times 2\times 2-\frac{1}{2}\times 9\times 9$

  $\begin{array}{l}=2-\frac{81}{2}\hfill \\ =\frac{4-81}{2}\hfill \\ =\frac{77}{2}\hfill \end{array}$