Question

mock exam for 2025 - 2026 - 1

1. calculate the definite integrals by referring to the following figure.

(1) $\int_{a}^{0}f(x)dx = 2.33$
(2) $\int_{a}^{c}f(x)dx = 8.34$
(3) $\int_{c}^{b}f(x)dx = 2.71$

Explanation:

Step1: Recall the property of definite integral and area

If \(y = f(x)\) is above the \(x -\)axis, \(\int_{m}^{n}f(x)dx\) is the area between \(y = f(x)\), \(x=m\), \(x = n\) and \(x-\)axis. If \(y = f(x)\) is below the \(x -\)axis, \(\int_{m}^{n}f(x)dx=- \) (the area between \(y = f(x)\), \(x=m\), \(x = n\) and \(x-\)axis)

Step2: Calculate \(\int_{a}^{0}f(x)dx\)

Since the function \(y = f(x)\) is below the \(x-\)axis when \(x\in[a,0]\), \(\int_{a}^{0}f(x)dx=-\text{Area}A\). Given \(\text{Area}A = 2.33\), so \(\int_{a}^{0}f(x)dx=- 2.33\)

Step3: Calculate \(\int_{a}^{c}f(x)dx\)

We know that \(\int_{a}^{c}f(x)dx=\int_{a}^{0}f(x)dx+\int_{0}^{b}f(x)dx+\int_{b}^{c}f(x)dx\). \(\int_{a}^{0}f(x)dx=-\text{Area}A=-2.33\), \(\int_{0}^{b}f(x)dx=\text{Area}B = 10.67\), \(\int_{b}^{c}f(x)dx=-\text{Area}C=-5.63\). Then \(\int_{a}^{c}f(x)dx=-2.33 + 10.67-5.63=2.71\)

Step4: Calculate \(\int_{c}^{b}f(x)dx\)

Since \(\int_{c}^{b}f(x)dx=-\int_{b}^{c}f(x)dx\), and \(\int_{b}^{c}f(x)dx=-\text{Area}C=-5.63\), so \(\int_{c}^{b}f(x)dx = 5.63\)

Answer:

(1) \(-2.33\)
(2) \(2.71\)
(3) \(5.63\)