Chapter 5.3, Problem 1E

(a)

To determine

To find: The value of integral.

Answer to Problem 1E

The value is 0.

Explanation of Solution

Given information: The expression is ${{\displaystyle \int }}_2^{2}g(x)dx$

  $f$ and $g$ are continuous that

  $\begin{array}{c}{{\displaystyle \int }}_1^{2}f(x)dx=-4,\\ {{\displaystyle \int }}_1^{5}f(x)dx=6,\\ {{\displaystyle \int }}_1^{5}g(x)dx=8\end{array}$

Calculation:

The given expression is ${{\displaystyle \int }}_2^{2}g(x)dx$

This integral is zero under the integration's definition. (See Table 5.3, Rule 2).

Therefore the required integral of the given expression is 0.

(b)

To determine

To find: The value integral.

Answer to Problem 1E

The value is $-8$ .

Explanation of Solution

Given information: The expression is ${{\displaystyle \int }}_5^{1}g(x)dx$

  $f$ and $g$ are continuous that

  $\begin{array}{c}{{\displaystyle \int }}_1^{2}f(x)dx=-4,\\ {{\displaystyle \int }}_1^{5}f(x)dx=6,\\ {{\displaystyle \int }}_1^{5}g(x)dx=8\end{array}$

Calculation:

The given expression is ${{\displaystyle \int }}_5^{1}g(x)dx$

The integral:

  $-{{\displaystyle \int }}_1^{5}g(x)dx=-8$

Therefore the required integral of the given expression is $-8$ .

(c)

To determine

To find: The value integral.

Answer to Problem 1E

The value is $-12$

Explanation of Solution

Given information: The expression is ${{\displaystyle \int }}_1^{2}3f(x)dx$

  $f$ and $g$ are continuous that

  $\begin{array}{c}{{\displaystyle \int }}_1^{2}f(x)dx=-4,\\ {{\displaystyle \int }}_1^{5}f(x)dx=6,\\ {{\displaystyle \int }}_1^{5}g(x)dx=8\end{array}$

Calculation:

The given expression is ${{\displaystyle \int }}_1^{2}3f(x)dx$

The integral:

  $\begin{array}{c}3{{\displaystyle \int }}_1^{2}f(x)dx=3\times (-4)\\ =-12\end{array}$

Therefore the required integral of the given expression is $-12$ .

(d)

To determine

To find: The value integral.

Answer to Problem 1E

The value is $10$

Explanation of Solution

Given information: The expression is ${{\displaystyle \int }}_2^{5}f(x)dx$

  $f$ and $g$ are continuous that

  $\begin{array}{c}{{\displaystyle \int }}_1^{2}f(x)dx=-4,\\ {{\displaystyle \int }}_1^{5}f(x)dx=6,\\ {{\displaystyle \int }}_1^{5}g(x)dx=8\end{array}$

Calculation:

The given expression is ${{\displaystyle \int }}_2^{5}f(x)dx$

The integral:

  ${{\displaystyle \int }}_2^{5}f(x)dx=-{{\displaystyle \int }}_1^{2}f(x)dx+{{\displaystyle \int }}_1^{5}f(x)dx$

  $\begin{array}{l}{{\displaystyle \int }}_2^{5}f(x)dx=4+6\hfill \\ =10\hfill \end{array}$

Therefore the required integral of the given expression ${{\displaystyle \int }}_2^{5}f(x)dx=10$

(e)

To determine

To find: The value integral.

Answer to Problem 1E

The value is $-2$ .

Explanation of Solution

Given information: The expression is ${{\displaystyle \int }}_1^5[f(x)-g(x)]dx$

  $f$ and $g$ are continuous that

  $\begin{array}{c}{{\displaystyle \int }}_1^{2}f(x)dx=-4,\\ {{\displaystyle \int }}_1^{5}f(x)dx=6,\\ {{\displaystyle \int }}_1^{5}g(x)dx=8\end{array}$

Calculation:

The given expression is ${{\displaystyle \int }}_1^5[f(x)-g(x)]dx$

The integral:

  $\begin{array}{c}{{\displaystyle \int }}_1^{5}f(x)dx-{{\displaystyle \int }}_1^{5}g(x)dx=6-8\\ =-2\end{array}$

Therefore the required integral of the given expression is $-2$ .

(f)

To determine

To find: The value integral.

Answer to Problem 1E

The value is $16$ .

Explanation of Solution

Given information: The expression is ${{\displaystyle \int }}_1^5[4f(x)-g(x)]dx$

  $f$ and $g$ are continuous that

  $\begin{array}{c}{{\displaystyle \int }}_1^{2}f(x)dx=-4,\\ {{\displaystyle \int }}_1^{5}f(x)dx=6,\\ {{\displaystyle \int }}_1^{5}g(x)dx=8\end{array}$

Calculation:

The given expression is ${{\displaystyle \int }}_1^5[4f(x)-g(x)]dx$

The integral:

  $\begin{array}{c}4{{\displaystyle \int }}_1^{5}f(x)dx-{{\displaystyle \int }}_1^{5}g(x)dx=4\times (6)-8\\ =24-8\\ =16\end{array}$

Therefore the required integral of the given expression is $16$ .