Applications of Integration: One Mark Questions, Book Back Answers and Notes

About Applications of Integration

Hard ~180 min study

The idea of integration is older than calculus. Archimedes found the area of a parabolic segment by filling it with smaller and smaller triangles, a technique called the method of exhaustion. Today we do the same thing with rectangles: cut the interval into thin strips, add their areas, and take the limit as the strips get thinner. That limit is the definite integral.

Computing such limits directly is slow, and the fundamental theorems of integral calculus give a shortcut: if you know an antiderivative $F$ of $f$, then $\int_a^b f(x)\,dx = F(b) - F(a)$. The unit then adds tools that make evaluation faster. Bernoulli's formula handles a polynomial times a function that is easy to integrate repeatedly. Reduction formulae deal with $\sin^n x$ and $\cos^n x$ between $0$ and $\pi/2$. The gamma integral evaluates integrals of the form $\int_0^\infty x^n e^{-ax}\,dx$ in one line.

The last two sections are the applications: the area of a region bounded by curves and lines, and the volume of a solid formed by revolving a region about an axis. For these, a neat sketch of the region is the first step. It shows you the limits of integration and which curve is on top. The same integrals are used in physics and engineering, for example to find work done by a variable force.

What you'll learn

• Evaluate a definite integral as the limit of a sum.
• Evaluate definite integrals using the fundamental theorems and properties.
• Apply reduction formulae to integrals of powers of sine and cosine.
• Evaluate improper integrals using the gamma integral.
• Use Bernoulli's formula to integrate products of a polynomial with another function.
• Find areas of bounded regions and volumes of solids of revolution.

Before you start

• Rules and techniques of differentiation.
• Indefinite integration, including standard integrals and integration by parts.
• Coordinate geometry and sketching standard curves such as lines, parabolas, circles and ellipses.

Applications of Integration explained

Applications of Integration: What to Know

The Definite Integral as a Limit of Sum

Divide $[a, b]$ into $n$ subintervals, choose a point in each, and add up $f(\text{point}) \times (\text{width})$. This is a Riemann sum. As $n \to \infty$ and the widths shrink to zero, the sum approaches $\int_a^b f(x) dx$. In the textbook exercises you evaluate simple integrals this way, using the formulas for $\sum r$ and $\sum r^2$. When $f(x) \ge 0$, the result is the area under the curve.

Fundamental Theorems and Properties

The first fundamental theorem says that $\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$. The second lets you evaluate a definite integral as $F(b) - F(a)$ for any antiderivative $F$. The properties save a lot of work. $\int_a^b f(x)\,dx = \int_a^b f(a + b - x)\,dx$ is the most used one. Over $[-a, a]$, an odd function integrates to zero and an even function gives twice the integral over $[0, a]$. When you substitute in a definite integral, change the limits too.

Bernoulli's Formula and Reduction Formulae

If $u$ is a polynomial, Bernoulli's formula gives $\int uv\,dx = uv_1 - u'v_2 + u''v_3 - \cdots$, where $v_1, v_2, v_3, \ldots$ are successive integrals of $v$. The signs alternate, and the series stops because some derivative of the polynomial becomes zero. For powers of sine and cosine, the reduction formula gives $\int_0^{\pi/2} \sin^n x\,dx = \frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdots$, ending with $\frac{1}{2} \cdot \frac{\pi}{2}$ when $n$ is even and with $\frac{2}{3}$ when $n$ is odd. The same result holds for $\cos^n x$.

Improper Integrals and the Gamma Integral

An integral whose interval is infinite, such as $\int_0^\infty$, is improper. It is defined as a limit, and it converges if that limit is finite. The gamma integral is $\Gamma(n) = \int_0^\infty e^{-x} x^{n-1}\,dx$ for $n > 0$. It satisfies $\Gamma(n + 1) = n\Gamma(n)$ and $\Gamma(n + 1) = n!$ for a positive integer $n$, so it extends the factorial. A result you will use often is $\int_0^\infty e^{-ax} x^n\,dx = \frac{n!}{a^{n+1}}$.

Area of Bounded Regions

The area between $y = f(x)$, the x-axis and the lines $x = a$ and $x = b$ is found with vertical strips. The area between $x = g(y)$, the y-axis and $y = c$, $y = d$ uses horizontal strips. For a region between two curves, integrate (upper curve minus lower curve), with the limits taken from their points of intersection. If a curve goes below the axis, the integral over that part is negative, so split the interval and add the absolute values. Always sketch first.

Volume of Solids of Revolution

When the region under $y = f(x)$ from $x = a$ to $x = b$ is revolved about the x-axis, each thin vertical strip sweeps out a disc. Adding these discs gives $V = \pi \int_a^b y^2 dx$. For revolution about the y-axis, use $V = \pi \int_c^d x^2\,dy$. These formulas can be used to derive standard volumes, such as those of a cone, a sphere and a frustum of a cone.

Common mistakes to avoid

• Forgetting to change the limits after a substitution in a definite integral.
• Taking the wrong curve as the upper boundary when finding the area between two curves.
• Using the x-axis volume formula with dx when the region is revolved about the y-axis.
• Integrating straight across a point where the curve crosses the x-axis, so positive and negative parts cancel and the area comes out too small.
• Using the even-power ending (with π/2) for an odd power in the reduction formula, or the other way round.

Test yourself on these with the practice test, then check the worked reasoning in the solved MCQs.

Frequently asked questions

How do I decide between vertical and horizontal strips?

Look at the boundaries. If the top and bottom of the region are given easily as functions of x, use vertical strips and integrate with respect to x. If the left and right boundaries are easier as functions of y, use horizontal strips. A sketch makes this clear.

Why is the gamma integral useful?

It evaluates integrals of a power of x times a decaying exponential over zero to infinity directly. For example, the integral of x³e^(−2x) from 0 to ∞ is 3!/2⁴ = 6/16 = 3/8. Without it you would need repeated integration by parts.

Why do some definite integrals come out as zero?

A common reason is an odd function integrated over a symmetric interval from −a to a, for example sin x or x³. The part below the x-axis cancels the part above it. Spotting this saves time in MCQs.

How does the fundamental theorem save work?

Instead of evaluating the limit of a sum, you find any antiderivative F of the integrand and compute F(b) − F(a). The theorem guarantees this equals the limit of the sum, which is why differentiation and integration are called inverse processes.

Is a definite integral always the same as the area?

No. A definite integral can be positive, negative or zero, because the parts below the x-axis count as negative. Area is always positive. If the curve crosses the axis, split the interval at the crossing point and add the absolute values of the pieces.

When should I use Bernoulli's formula instead of integration by parts?

Use it when one factor is a polynomial and the other can be integrated repeatedly, such as eˣ, sin x or cos x. It is integration by parts done all at once, so you write one line with alternating signs instead of several steps.