average value of a function

(or f_ave) the average value of a function on an interval can be found by calculating the definite integral of the function and dividing that value by the length of the interval

a primary operation of calculus; the area between the curve and the x-axis over a given interval is a definite integral

a function is integrable if the limit defining the integral exists; in other words, if the limit of the Riemann sums as n goes to infinity exists

the function to the right of the integration symbol; the integrand includes the function being integrated

limits of integration

these values appear near the top and bottom of the integral sign and define the interval over which the function should be integrated

the area between a function and the x-axis such that the area below the x-axis is subtracted from the area above the x-axis; the result is the same as the definite integral of the function

total area between a function and the x-axis is calculated by adding the area above the x-axis and the area below the x-axis; the result is the same as the definite integral of the absolute value of the function

variable of integration

indicates which variable you are integrating with respect to; if it is x, then the function in the integrand is followed by dx

5.2 The Definite Integral: practice

In the following exercises, express the limits as integrals.

$lim_{n \to \infty} \sum_{i = 1}^{n} (5 (x_{i}^{*})^{2} - 3 (x_{i}^{*})^{3}) Δ x$ over

In the following exercises, express the limits as integrals.

$lim_{n \to \infty} \sum_{i = 1}^{n} (5 (x_{i}^{*})^{2} - 3 (x_{i}^{*})^{3}) Δ x$ over

In the following exercises, express the limits as integrals.

$lim_{n \to \infty} \sum_{i = 1}^{n} cos^{2} (2 π x_{i}^{*}) Δ x$ over $[0, 1]$

In the following exercises, given L $_{n}$ or R $_{n}$ as indicated, express their limits as $n \to \infty$ as definite integrals, identifying the correct intervals.

$R_{n} = \frac{1}{n} \sum_{i = 1}^{n} \frac{i}{n}$

In the following exercises, given L $_{n}$ or R $_{n}$ as indicated, express their limits as $n \to \infty$ as definite integrals, identifying the correct intervals.

$R_{n} = \frac{3}{n} \sum_{i = 1}^{n} (3 + 3 \frac{i}{n})$

In the following exercises, given L $_{n}$ or R $_{n}$ as indicated, express their limits as $n \to \infty$ as definite integrals, identifying the correct intervals.

$R_{n} = \frac{1}{n} \sum_{i = 1}^{n} (1 + \frac{i}{n}) lo g ((1 + \frac{i}{n})^{2})$

In the following exercises, evaluate the integrals of the functions graphed using the formulas for areas of triangles and circles, and subtracting the areas below the x-axis.

A graph of three isosceles triangles corresponding to the functions 1 - |x-1| over [0,2], 2 - |x-4| over [2,4], and 3 - |x-9| over [6,12]. The first triangle has endpoints at (0,0), (2,0), and (1,1). The second triangle has endpoints at (2,0), (6,0), and (4,2). The last has endpoints at (6,0), (12,0), and (9,3). All three are shaded.

In the following exercises, evaluate the integrals of the functions graphed using the formulas for areas of triangles and circles, and subtracting the areas below the x-axis.

A graph of three shaded triangles. The first has endpoints at (0, 0), (2, 0), and (1, 1) and corresponds to the function 1 - |x-1| over [0, 2]. The second has endpoints at (2, 0), (6, 0), and (4, -2) and corresponds to the function |x-4| - 2 over [2, 6]. The third has endpoints at (6, 0), (12, 0), and (9, 3) and corresponds to the function 3 - |x-9| over [6, 12].

In the following exercises, evaluate the integrals of the functions graphed using the formulas for areas of triangles and circles, and subtracting the areas below the x-axis.

A graph with three shaded parts. The first is a triangle with endpoints at (0, 0), (2, 0), and (1, 1), which corresponds to the function 1 - |x-1| over [0, 2] in quadrant 1. The second is the lower half of a circle with center at (4, 0) and radius two, which corresponds to the function –sqrt(-12 + 8x – x^2) over [2, 6]. The last is a triangle with endpoints at (6, 0), (12, 0), and (9, 3), which corresponds to the function 3 - |x-9| over [6, 12].

In the following exercises, evaluate the integral using area formulas.

$\int_{0}^{3} (3 - x) d x$

In the following exercises, evaluate the integral using area formulas.

$\int_{2}^{3} (3 - x) d x$

In the following exercises, evaluate the integral using area formulas.

$\int_{0}^{6} (3 - ∣ x - 3∣) d x$

In the following exercises, evaluate the integral using area formulas.

$\int_{1}^{5} 4 - (x - 3)^{2} d x$

In the following exercises, evaluate the integral using area formulas.

$\int_{- 2}^{3} (3 - ∣ x ∣) d x$

In the following exercises, use averages of values at the left (L) and right (R) endpoints to compute the integrals of the piecewise linear functions with graphs that pass through the given list of points over the indicated intervals.

${(0, 2), (1, 0), (3, 5), (5, 5), (6, 2), (8, 0)}$ over $[0, 8]$

${(- 4, 0), (- 2, 2), (0, 0), (1, 2), (3, 2), (4, 0)}$ over $[- 4, 4]$

Suppose that $\int_{0}^{4} f (x) d x = 5$ and $\int_{0}^{2} f (x) d x = - 3,$ and $\int_{0}^{4} g (x) d x = - 1$ and $\int_{0}^{2} g (x) d x = 2.$ In the following exercises, compute the integrals.

$\int_{2}^{4} (f (x) + g (x)) d x$

Suppose that $\int_{0}^{4} f (x) d x = 5$ and $\int_{0}^{2} f (x) d x = - 3,$ and $\int_{0}^{4} g (x) d x = - 1$ and $\int_{0}^{2} g (x) d x = 2.$ In the following exercises, compute the integrals.

$\int_{2}^{4} (f (x) - g (x)) d x$

Suppose that $\int_{0}^{4} f (x) d x = 5$ and $\int_{0}^{2} f (x) d x = - 3,$ and $\int_{0}^{4} g (x) d x = - 1$ and $\int_{0}^{2} g (x) d x = 2.$ In the following exercises, compute the integrals.

$\int_{2}^{4} (4 f (x) - 3 g (x)) d x$

In the following exercises, use the identity $\int_{- A}^{A} f (x) d x = \int_{- A}^{0} f (x) d x + \int_{0}^{A} f (x) d x$ to compute the integrals.

$\int_{- π}^{π} \frac{t}{1 + c o s t} d t$

In the following exercises, find the net signed area between $f (x)$ and the x-axis.

$\int_{2}^{4} (x - 3)^{3} d x$ (Hint: Look at the graph of f.)

In the following exercises, given that $\int_{0}^{1} x d x = \frac{1}{2}, \int_{0}^{1} x^{2} d x = \frac{1}{3},$ and $\int_{0}^{1} x^{3} d x = \frac{1}{4},$ compute the integrals.

$\int_{0}^{1} (1 - x + x^{2} - x^{3}) d x$

In the following exercises, given that $\int_{0}^{1} x d x = \frac{1}{2}, \int_{0}^{1} x^{2} d x = \frac{1}{3},$ and $\int_{0}^{1} x^{3} d x = \frac{1}{4},$ compute the integrals.

$\int_{0}^{1} (1 - 2 x)^{3} d x$

In the following exercises, given that $\int_{0}^{1} x d x = \frac{1}{2}, \int_{0}^{1} x^{2} d x = \frac{1}{3},$ and $\int_{0}^{1} x^{3} d x = \frac{1}{4},$ compute the integrals.

$\int_{0}^{1} (7 - 5 x^{3}) d x$

In the following exercises, use the comparison theorem.

Show that $\int_{- 2}^{3} (x - 3) (x + 2) d x \leq 0.$

In the following exercises, use the comparison theorem.

Show that $\int_{1}^{2} 1 + x d x \leq \int_{1}^{2} 1 + x^{2} d x .$

In the following exercises, use the comparison theorem.

Show that $\int_{- π / 4}^{π / 4} cos t d t \geq π 2 / 4.$

In the following exercises, find the average value f $_{a v e}$ of f between a and b, and find a point c, where $f (c) = f_{ave} .$

$f (x) = x^{5}, a = - 1, b = 1$

In the following exercises, find the average value f $_{a v e}$ of f between a and b, and find a point c, where $f (c) = f_{ave} .$

$f (x) = (3 - ∣ x ∣), a = - 3, b = 3$

In the following exercises, find the average value f $_{a v e}$ of f between a and b, and find a point c, where $f (c) = f_{ave} .$

$f (x) = cos x, a = 0, b = 2 π$

In the following exercises, approximate the average value using Riemann sums L₁₀₀ and R₁₀₀. How does your answer compare with the exact given answer?

[T] $y = e^{x / 2}$ over the interval $[0, 1];$ the exact solution is $2 (e - 1) .$

In the following exercises, approximate the average value using Riemann sums L₁₀₀ and R₁₀₀. How does your answer compare with the exact given answer?

[T] $y = \frac{x + 1}{4 - x ^{2}}$ over the interval $[- 1, 1];$ the exact solution is $\frac{π}{6} .$