Activity 55.
Suppose that the following information is known about the functions \(f\text{,}\) \(g\text{,}\) \(x^2\text{,}\) and \(x^3\text{:}\)
-
\(\int_0^2 f(x) \, dx = -3\text{;}\) \(\int_2^5 f(x) \, dx = 2\)
-
\(\int_0^2 g(x) \, dx = 4\text{;}\) \(\int_2^5 g(x) \, dx = -1\)
-
\(\int_0^2 x^2 \, dx = \frac{8}{3}\text{;}\) \(\int_2^5 x^2 \, dx = \frac{117}{3}\)
-
\(\int_0^2 x^3 \, dx = 4\text{;}\) \(\int_2^5 x^3 \, dx = \frac{609}{4}\)
Use the provided information and the rules discussed in the preceding section to evaluate each of the following definite integrals.
(a)
\(\int_5^2 f(x) \, dx\)
Hint.
Note that the value of \(\int_2^5 f(x) \, dx\) is given.
Answer.
\(\int_5^2 f(x) \,dx = -2\text{.}\)
Solution.
Note that the value of \(\int_2^5 f(x) \, dx\) is given, and thus
\begin{equation*}
\int_5^2 f(x) \,dx = -\int_2^5 f(x) \, dx = -2\text{.}
\end{equation*}
(b)
\(\int_0^5 g(x) \, dx\)
Hint.
Use the values of \(\int_0^2 g(x) \,dx\) and \(\int_2^5 g(x) \,dx\text{.}\)
Answer.
\(\int_0^5 g(x) \,dx = 3\text{.}\)
Solution.
Since \(\int_0^2 g(x) \,dx = 4\) and \(\int_2^5 g(x) \,dx = -1\text{,}\) we have
\begin{equation*}
\int_0^5 g(x) \,dx = \int_0^2 g(x) \,dx + \int_2^5 g(x) \,dx = 4 + (-1) = 3\text{.}
\end{equation*}
(c)
\(\int_0^5 (f(x) + g(x))\, dx\)
Hint.
First find \(\int_0^5 f(x) \, dx\) and \(\int_0^5 g(x) \, dx\text{.}\)
Answer.
\(\int_0^5 (f(x) + g(x))\, dx = 2\text{.}\)
Solution.
First, using work from and similar to that in (c), we find \(\int_0^5 f(x) \, dx = -3 + 2 = -1\) and \(\int_0^5 g(x) \, dx = 3\text{,}\) thus by the sum rule,
\begin{equation*}
\int_0^5 (f(x) + g(x))\, dx = \int_0^5 f(x)\, dx + \int_0^5 g(x)\, dx = -1 + 3 = 2\text{.}
\end{equation*}
(d)
\(\int_2^5 (3x^2 - 4x^3) \, dx\)
Hint.
Use the sum and constant multiple rules.
Answer.
\(\int_2^5 (3x^2 - 4x^3) \, dx = -492\text{.}\)
Solution.
By the sum and constant multiple rules,
\begin{equation*}
\int_2^5 (3x^2 - 4x^3) \, dx = 3\int_2^5 x^2 \, dx - 4\int_2^5 x^3 \, dx = 3 \cdot \frac{117}{3} - 4 \frac{609}{4} = 117 - 609 = -492\text{.}
\end{equation*}
(e)
\(\int_5^0 (2x^3 - 7g(x)) \, dx\)
Hint.
First write \(\int_5^0 (2x^3 - 7g(x)) \, dx = -\int_0^5 (2x^3 - 7g(x)) \, dx\text{.}\)
Answer.
\(\int_5^0 (2x^3 - 7g(x)) \, dx = -\frac{583}{2}\text{.}\)
Solution.
First, we write \(\int_5^0 (2x^3 - 7g(x)) \, dx = -\int_0^5 (2x^3 - 7g(x)) \, dx\text{.}\) Then, using the sum and constant multiple rules, it follows
\begin{align*}
\int_5^0 (2x^3 - 7g(x)) \, dx =\mathstrut \amp -\int_0^5 (2x^3 - 7g(x)) \, dx\\
=\mathstrut \amp -\left(2 \int_0^5 x^3 \, dx - 7 \int_0^5 g(x) \,dx \right)\\
=\mathstrut \amp -2 \left(4 + \frac{609}{4}\right) + 7 \left(4 + (-1))\right)\\
=\mathstrut \amp -\frac{625}{2} + 21\\
=\mathstrut \amp -\frac{583}{2}\text{.}
\end{align*}
