Question Statement

Using the area shown below in the figure, evaluate the integrals:

(i) ${\int }_a^{b}f(x)dx$

(ii) ${\int }_b^{c}f(x)dx$

(iii) ${\int }_c^{d}f(x)dx$

(iv) ${\int }_a^{c}f(x)dx$

(v) ${\int }_b^{d}f(x)dx$

(vi) ${\int }_a^{d}f(x)dx$

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Background and Explanation

This problem demonstrates the geometric interpretation of definite integrals as the area under a curve. When $f(x)\ge 0$ on an interval $[a,b]$, the definite integral ${\int }_a^{b}f(x)dx$ represents the area bounded by the curve $y=f(x)$, the x-axis, and the vertical lines $x=a$ and $x=b$. Additionally, definite integrals satisfy the additive property over adjacent intervals.

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Solution

Part (i): ${\int }_a^{b}f(x)dx$

From the figure, it is clear that the area between $a$ to $b$ is $A_1=2$. Since this region lies above the x-axis, the definite integral equals this geometric area:

${\int }_a^{b}f(x)dx=A_1=2$

Part (ii): ${\int }_b^{c}f(x)dx$

From the figure, it is clear that the area between $b$ to $c$ is $A_2=3$. Therefore:

${\int }_b^{c}f(x)dx=A_2=3$

Part (iii): ${\int }_c^{d}f(x)dx$

From the figure, it is clear that the area from $c$ to $d$ is $A_3=3.5$. Therefore:

${\int }_c^{d}f(x)dx=A_3=3.5$

Part (iv): ${\int }_a^{c}f(x)dx$

To find the integral over the combined interval $[a,c]$, we apply the additive property of definite integrals by splitting the interval at point $b$:

$\begin{array}{rl}{\int }_a^{c}f(x)dx& ={\int }_a^{b}f(x)dx+{\int }_b^{c}f(x)dx\\ & =2+(3)(\text{ Using part (i) \& (ii)) }\\ & =2+3\\ & =5\end{array}$

Part (v): ${\int }_b^{d}f(x)dx$

Similarly, we split the interval $[b,d]$ at point $c$ and sum the areas from parts (ii) and (iii):

$\begin{array}{rl}{\int }_b^{d}f(x)dx& ={\int }_b^{c}f(x)dx+{\int }_c^{d}f(x)dx\\ & =3+3.5\\ & =6.5\end{array}$

Part (vi): ${\int }_a^{d}f(x)dx$

For the entire interval $[a,d]$, we decompose the integral into the sum of all three subintervals:

$\begin{array}{rl}{\int }_a^{d}f(x)dx& ={\int }_a^{b}f(x)dx+{\int }_b^{c}f(x)dx+{\int }_c^{d}f(x)dx\\ & =2+(3)+3.5\\ & =8.5\end{array}$

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Key Formulas or Methods Used

• Geometric interpretation of definite integrals: ${\int }_a^{b}f(x)dx=\text{Area under curve from }a\text{ to }b$ (when $f(x)\ge 0$)
• Additive property of definite integrals: ${\int }_a^{c}f(x)dx={\int }_a^{b}f(x)dx+{\int }_b^{c}f(x)dx$ for $a<b<c$
• Interval decomposition: Breaking larger intervals into smaller adjacent subintervals to compute total area

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Summary of Steps

1. Identify the three individual areas from the figure: $A_1=2$ (interval $[a,b]$), $A_2=3$ (interval $[b,c]$), and $A_3=3.5$ (interval $[c,d]$).
2. For single-interval integrals (parts i-iii), assign the corresponding area value directly to the integral.
3. For combined intervals (parts iv-vi), apply the additive property by splitting the integral at intermediate points ($b$ or $c$).
4. Substitute the known area values and sum them algebraically to obtain the final results (5, 6.5, and 8.5 respectively).