Let $f(t)=F'(t).$ Write the integral $\int_{a}^{b}f(t)\ dt$ and evaluate it using the Fundamental Theorem of Calculus.

$$F(t) = t^{4}, a = −1, b = 1$$

Find the average value of the function over the given interval.

$f(x) = 2$ over $[a, b]$

Use a calculator or a computer to find the value of the definite integral.

$$\displaystyle \int_{0}^{1}\sin(t^{2}) \ dt$$

Your velocity is $v(t)=\sin \left(t^2\right) \mathrm{mph}$ for $0 \leq t \leq 1.1$. Estimate the distance traveled during this time.

Which of the following statements follow directly from the rule

$$\displaystyle \int_{a}^{b}(f(x)+g(x))\ dx= \displaystyle \int_{a}^{b}f(x)\ dx + \displaystyle \int_{a}^{b}g(x)\ dx?$$

(a) If $\int_{a}^{b}(f(x)+g(x))\ dx=5+7,$ then $\int_{a}^{b}f(x)\ dx=5$ and $\int_{a}^{b}g(x)\ dx=7.$

For the even function f graphed in the given figure:

Suppose you know $\int_2^5 f(x) d x$ and $\int_{-2}^5 f(x) d x$. What is $\int_0^2 f(x) d x$ ?

The average value of the product, $f(x)g(x)$, of two functions on an interval equals the product of the average values of $f(x)$ and $g(x)$ on the interval.

Let $f(t)=F'(t).$ Write the integral $\int_{a}^{b}f(t) \ dt$ and evaluate it using the Fundamental Theorem of Calculus.

$$F(t) = t^{2}; a = 1, b = 3$$

On the interval $[a, b],$ the average value of $f(x)+g(x)$ is the average value of $f(x)$ plus the average value of $g(x).$

For the even function f graphed in the given figure:

Suppose you know $\int_{-2}^5 f(x) d x$ and $\int_{-2}^0 f(x) d x$. What is $\int_2^5 f(x) d x$ ?

Explain in words what the integral represents and give units. $\int_{0}^{6} a(t) d t$, where a(t) is acceleration in $\mathrm{km} / \mathrm{hr}^{2}$ and t is time in hours.

Suppose that $A$ is the average value of $f$ on the interval $[1, 4]$ and $B$ is the average value of $f$ on the interval $[4, 9].$ Then the average value of $f$ on $[1, 9]$ is the weighted average $(3/8)A + (5/8)B.$

For the even function f graphed in the given figure:

Suppose you know $\int_{-2}^2 f(x) d x$ and $\int_0^5 f(x) d x$. What is $\int_2^5 f(x) d x$ ?

If $a < c < d < b,$ then the average value of $f$ on the interval $[c, d]$ is less than the average value of $f$ on the interval $[a, b].$