Exercise Questions

All questions in this exercise are listed below. Click on a question to view its solution.

Key Concepts

This exercise focuses on the following concepts:

• Definite integrals as signed area under curves

• Geometric interpretation of definite integrals (rectangles, triangles, circles)

• Properties of definite integrals (linearity and additivity)

• Integration of piecewise-defined functions

• Evaluating integrals using geometric area formulas

• Interval additivity property: ${\int }_a^{c}f(x)dx={\int }_a^{b}f(x)dx+{\int }_b^{c}f(x)dx$

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Important Formulas

Signed Area and the Definite Integral

The definite integral ${\int }_a^{b}f(x)dx$ represents the net signed area between the curve $y=f(x)$ and the $x$-axis over $[a,b]$:

• Regions above the $x$-axis contribute positive area.
• Regions below the $x$-axis contribute negative area.

When $f(x)\ge 0$ on $[a,b]$, the definite integral equals the actual geometric area of the region bounded by $y=f(x)$, the $x$-axis, and the lines $x=a$ and $x=b$.

Properties of Definite Integrals

Zero-width interval: ${\int }_a^{a}f(x)dx=0$

Reversing limits: ${\int }_b^{a}f(x)dx=-{\int }_a^{b}f(x)dx$

Linearity Property: ${\int }_a^b\left[c_{1}f(x)+c_{2}g(x)\right]dx=c_1{\int }_a^{b}f(x)dx+c_2{\int }_a^{b}g(x)dx$

Interval Additivity: ${\int }_a^{c}f(x)dx={\int }_a^{b}f(x)dx+{\int }_b^{c}f(x)dx$

Geometric Area Formulas

When the integrand defines a standard geometric shape, the integral can be evaluated directly using the corresponding area formula:

Example: ${\int }_0^r\sqrt{r^2-x^2}dx=\frac{1}{4}\pi r^2$ because the integrand $y=\sqrt{r^2-x^2}$ over $[0,r]$ traces a quarter-circle of radius $r$.

Piecewise Functions

For a piecewise function $f(x)$ whose definition changes at $x=k$ within $[a,b]$, use interval additivity to split the integral:

${\int }_a^{b}f(x)dx={\int }_a^k{f}_1(x)dx+{\int }_k^b{f}_2(x)dx$

where $f_1$ and $f_2$ are the respective pieces. Each sub-integral is then evaluated using the appropriate formula for that piece.

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Summary

This exercise develops geometric and algebraic techniques for evaluating definite integrals without using antiderivatives. Key strategies include:

1. Visualizing signed area — recognizing whether regions lie above or below the $x$-axis.
2. Geometric shortcuts — identifying triangles, rectangles, and semicircles within integral bounds and applying their area formulas directly.
3. Linearity — splitting sums/differences of integrands and factoring out constants.
4. Interval additivity — splitting integrals at boundary points, especially for piecewise functions.
5. Limit reversal — understanding that swapping limits negates the integral value.