Problem 165 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The area enclosed by the curve \(x=a \cos ^{3} t, y=b \sin ^{3}\) \(t, 0 \leq t \leq 2 \pi\) is (A) \(\frac{3 \pi a b}{8}\) (B) \(\frac{3 \pi a b}{4}\) (C) \(\frac{3 a b}{8}\) (D) None of these

Step-by-Step Solution

Answer

The area is \(\frac{3 \pi a b}{8}\), corresponding to option A.

Step 1: Identify the Shape and Parametric Form

The given parametric equations are \(x = a \cos^3 t\) and \(y = b \sin^3 t\). These describe a bean-shaped closed curve called the astroid, but squashed or stretched based on \(a\) and \(b\).

Step 2: Setup the Integral for Area

The area enclosed by a parametric curve \((x(t), y(t))\) over an interval \([\alpha, \beta]\) can be found using the formula: \(A = \int_{\alpha}^{\beta} y(t) x'(t) \, dt\). For this curve: \(x'(t) = \frac{d}{dt}(a \cos^3 t) = -3a \cos^2 t \sin t\).

Step 3: Substitute and Simplify the Integral

Substitute the expressions for \(x'(t)\) and \(y(t)\) into the area integral: \(A = \int_0^{2\pi} b \sin^3 t \cdot (-3a \cos^2 t \sin t) \, dt\). Simplify to get: \(A = -3ab \int_0^{2\pi} \sin^4 t \cos^2 t \, dt\).

Step 4: Use Trigonometric Identities for Reduction

We use \(\sin^4 t = (1 - \cos^2 t)^2\) and \(\cos^2 t\). The integral becomes: \(A = -3ab \int_0^{2\pi} (1 - 2\cos^2 t + \cos^4 t) \cos^2 t \, dt\).

Step 5: Simplify Further and Solve the Integral

This expands to \(-3ab \int_0^{2\pi} (\cos^2 t - 2\cos^4 t + \cos^6 t) \, dt\). These integrals can be solved by simple substitution or known results for powers of trig functions over \(0\) to \(2\pi\).

Step 6: Calculate Simplified Integral

Using known results, we find that \(\int_0^{2\pi} \cos^{2n} t \, dt = \frac{1}{2^{n-1}} \cdot \frac{(2n-1)!!}{(2n)!!} \cdot 2\pi\). Specifically: \(\int_0^{2\pi} \cos^2 t \, dt = \pi, \int_0^{2\pi} \cos^4 t \, dt = \frac{3\pi}{4}, \int_0^{2\pi} \cos^6 t \, dt = \frac{5\pi}{8}\).

Step 7: Substitute Known Values and Solve for Area

Substitute these values back into the integral to find: \(-3ab(\pi - 2\cdot\frac{3\pi}{4} + \frac{5\pi}{8})\). Calculate this to get: \(-3ab(\pi - \frac{3\pi}{2} + \frac{5\pi}{8}) = \frac{3\pi ab}{8}\).

Step 8: Interpret the Negative Sign

Since the integral calculates the area through a clockwise direction due to the negative sign, the actual area is positive, thus: \(\frac{3\pi ab}{8}\). Match this with the options: Option A is correct.

Key Concepts

Astroid

An astroid is a type of curve that looks like a star with four "cusps" or pointed ends. It falls under a class of shapes known as hypocycloids. In terms of parametric equations, an astroid is represented as:

• For a standard astroid: \(x = a \cos^3 t\) and \(y = a \sin^3 t\), where \(a\) is a constant that controls the size of the astroid.
• The curve arises when a smaller circle rolls on the inside of a larger circle. The path traced by a point on the smaller circle forms the astroid.

This particular problem modifies the astroid slightly with distinct scalars for \(x\) and \(y\), \(a\) and \(b\) respectively. This transforms it from the perfect star shape to an elongated or compressed version depending on the ratio \(a/b\). Astroids have practical applications in gear systems and optical design due to their unique properties.

Area under a curve

The area under a curve represents how much space it occupies on a graph, typically within certain bounds. For parametric equations like the ones used with the astroid, calculating the area involves an integral. The key formula is:

• \(A = \int_{\alpha}^{\beta} y(t) x'(t) \, dt\), where \(x(t)\) and \(y(t)\) are parametric functions.
• In this exercise, \(x(t) = a \cos^3 t\) and \(y(t) = b \sin^3 t\), making \(x'(t) = -3a \cos^2 t \sin t\).
• The integral evaluates the combined effect of \(x'\) and \(y\) over one complete cycle (from \(0\) to \(2\pi\)).

The process translates the geometric space within the curve into a numerical value, using calculus to handle curves that aren’t straight lines. The integral's result turns into the actual area occupied by the astroid-like shape on the plane.

Trigonometric identities

Trigonometric identities are equations involving trigonometric functions that are universally true. They are handy for simplifying expressions and solving integrals, especially with geometric problems involving curves.

• In the problem, the identity \(\sin^4 t = (1 - \cos^2 t)^2\) is used to simplify the integral for the area.
• These identities allow the integral \(-3ab \int_0^{2\pi} \sin^4 t \cos^2 t \, dt\) to be reduced to solvable forms.
• Another identity, \(\cos^2 t = \frac{1}{2}(1 + \cos 2t)\), simplifies calculations when integrating powers of cosines.

Integrating terms like \(\cos^2 t\), \(\cos^4 t\), and \(\cos^6 t\) rely on these identities to convert trigonometric powers into manageable forms. They are crucial in calculating the total area by converting complex trigonometric forms into sums of simpler integral terms.