Definite Integrals I

1

Area Interpretation Solution

The graph of \(f\) is shown. Evaluate each integral by interpreting it in terms of areas.

(a) \(\displaystyle\int_{0}^{4} f(x)\,dx\) (b) \(\displaystyle\int_{6}^{10} f(x)\,dx\) (c) \(\displaystyle\int_{0}^{10} f(x)\,dx\)

2

Lines & Semicircle Solution

The graph of \(g\) consists of two straight lines and a semicircle. Use it to evaluate each integral.

(a) \(\displaystyle\int_{0}^{4} g(x)\,dx\) (b) \(\displaystyle\int_{0}^{8} g(x)\,dx\) (c) \(\displaystyle\int_{8}^{16} g(x)\,dx\) (d) \(\displaystyle\int_{0}^{18} g(x)\,dx\)

3

Curve & Line Graph Solution

Consider the graph of \(y = f(x)\), which is a curve for \(x \leq 0\) and a line for \(x \geq 0\). Given that the area of the left region is 6.5 as shown, evaluate each integral.

(a) \(\displaystyle\int_{0}^{-3} f(x)\,dx\) (b) \(\displaystyle\int_{0}^{4} f(x)\,dx\) (c) \(\displaystyle\int_{0}^{8} f(x)\,dx\)

4

Absolute Value & Linear Solution

Evaluate each of the following integrals by interpreting it in terms of areas.

(a) \(\displaystyle\int_{-2}^{3} |x|\,dx\) (b) \(\displaystyle\int_{-1}^{4} (4-2x)\,dx\)

5

Shifted Absolute & Semicircle Solution

Evaluate each of the following integrals by interpreting it in terms of areas.

(a) \(\displaystyle\int_{0}^{6} |x-4|\,dx\) (b) \(\displaystyle\int_{-2}^{2} \sqrt{4-x^2}\,dx\)

6

Absolute Value & Semicircle Solution

Evaluate each integral by interpreting it in terms of areas.

(a) \(\displaystyle\int_{-2}^{1} |x|\,dx\) (b) \(\displaystyle\int_{1}^{4} |x-2|\,dx\) (c) \(\displaystyle\int_{-2}^{2} \sqrt{4-t^2}\,dt\)

7

Additive Interval Property Solution

If \(\displaystyle\int_0^5 f(x)\,dx = 6\) and \(\displaystyle\int_3^5 f(x)\,dx = 2\), find \(\displaystyle\int_0^3 f(x)\,dx\).

8

Linearity of Integrals Solution

If \(\displaystyle\int_1^6 f(x)\,dx = 7\) and \(\displaystyle\int_1^6 g(x)\,dx = -2\), find:

(a)

\(\displaystyle\int_1^6 [3f(x) - 2g(x)]\,dx\)

(b)

\(\displaystyle\int_1^6 [3 + f(x)]\,dx\)