Chapter 5.2, Problem 51E

(a)

To determine

To find: The points of discontinuity of the integrand on the interval of integration.

Answer to Problem 51E

There is a discontinuity at $x=-1$ .

Explanation of Solution

Given information:

The integration is: ${{\displaystyle \int }}_{-3}^4\frac{x^2-1}{x+1}dx$

Graph:

The graph of the function $y=\frac{x^2-1}{x+1}$ on the interval $[-3,4]$ .

  

While the graph doesn't appear to be discontinuous, the function is discontinuous $x=-1$

Hence substituting in $x=-1$ into

  $y=\frac{x^2-1}{x+1}$ would have division by $0$ .

The required function is discontinuous at $x=-1$

(b)

To determine

To find: The integral.

Answer to Problem 51E

The integral $-\frac{7}{2}$

Explanation of Solution

Given information:

The integration is: ${{\displaystyle \int }}_{-3}^4\frac{x^2-1}{x+1}dx$

Calculation:

The graph of the function $y=\frac{x^2-1}{x+1}$ on the interval $[-3,4]$ .

The integral:

  $-\frac{1}{2}\times 4\times 4+\frac{1}{2}\times 3\times 3=\frac{-7}{2}$

The triangle between 1 and 4 has a base of 3 and a height of 3, but the triangle between $\text{-3}$ and 1 has a base of 4 and a height of 4, and its area must be negative because it lies below the $x$ -axis.

  

Therefore the required integral of the given integration ${{\displaystyle \int }}_{-3}^4\frac{x^2-1}{x+1}dx$ is $-\frac{7}{2}$ .