Mixed Problems | JEE Main Calculus Crash Course | Mathematicon

Mixed Problems: Mastering Area Under Curves for JEE Main

Welcome, JEE aspirants! This lesson, "Mixed Problems," is where we bring together everything you've learned about area under curves. You'll see how to tackle complex scenarios involving multiple curves, parametric equations, and those tricky JEE-style application problems. Buckle up, because this is where theory meets serious problem-solving!

Area Bounded by Multiple Curves

Often, you'll encounter regions bounded not by a single curve and the axes, but by the intersection of several curves. The key here is to break down the region into smaller, manageable parts. Identify the points of intersection and determine which curve is "on top" within each interval.

Conceptual Explanation: Imagine you're finding the area of a lake with an irregular shoreline. You wouldn't try to calculate the whole thing at once. Instead, you'd divide it into sections, measure each section, and then add them up. That's precisely what we do with multiple curves!

Example: Consider finding the area bounded by $y = x^2$ and $y = 2x$ . First, find their intersection points: $x^2 = 2x \Rightarrow x = 0, 2$ . Then, the area is given by integrating the difference between the "top" curve ( $y=2x$ ) and the "bottom" curve ( $y=x^2$ ) from $x=0$ to $x=2$ .

Area Using Parametric Equations

Sometimes, curves are defined using parametric equations, where both $x$ and $y$ are expressed in terms of a parameter (usually $t$ or $\theta$ ). In such cases, we need to adapt our integration techniques.

Area = \int_{t_1}^{t_2} y(t) \cdot x'(t) \, dt

Explanation: This formula transforms the integral from $dy$ to $dt$ . $y(t)$ represents the y-coordinate as a function of the parameter $t$ , and $x'(t)$ is the derivative of $x$ with respect to $t$ , which essentially replaces $dx$ . The limits of integration, $t_1$ and $t_2$ , are the parameter values corresponding to the start and end points of the curve.

Example: Find the area under the curve defined by $x = a\cos\theta$ , $y = b\sin\theta$ , where $0 \le \theta \le \pi/2$ . Here, $x'(\theta) = -a\sin\theta$ . The area is then $\int_{0}^{\pi/2} (b\sin\theta)(-a\sin\theta) \, d\theta$ . Remember to take the absolute value since area cannot be negative.

JEE-Style Application Problems

JEE problems often combine area under curves with other concepts like limits, derivatives, and trigonometry. Expect questions that require you to think outside the box and apply multiple ideas.

Example: A question might involve finding the area enclosed by a curve whose equation is implicitly defined (e.g., $x^3 + y^3 = 3axy$ ) or finding the area bounded by the locus of a point satisfying certain geometric conditions.

Strategy for Complex Regions

Sketch the curves: A clear diagram is half the battle won. Always sketch the curves to visualize the region.
Find intersection points: Solve the equations simultaneously to find where the curves intersect. These points define the limits of integration.
Determine the "top" and "bottom" curves: Within each interval, identify which curve has the higher y-value. This determines which function to subtract from which.
Set up the integral(s): Divide the region into sub-regions if necessary. Set up the definite integral(s) representing the area of each sub-region.
Evaluate the integral(s): Use your integration skills to find the area of each sub-region.
Add the areas: Sum up the areas of all sub-regions to get the total area.

Tip: Look for symmetry! If the region is symmetric about the x-axis or y-axis, you can calculate the area of one half and double it to get the total area. This can significantly simplify the integration process.

Important Formulas: Revisited

Area between curves $f(x)$ and $g(x)$ from $a$ to $b$ , where $f(x) \ge g(x)$ : $Area = \int_{a}^{b} [f(x) - g(x)] \, dx$

Area using parametric equations $x = x(t)$ , $y = y(t)$ from $t_1$ to $t_2$ : $Area = \left| \int_{t_1}^{t_2} y(t) \cdot x'(t) \, dt \right|$

Area bounded by a polar curve $r = f(\theta)$ from $\theta_1$ to $\theta_2$ : $Area = \frac{1}{2} \int_{\theta_1}^{\theta_2} [f(\theta)]^2 \, d\theta$

Trick: Remember to use the correct formula based on how the curve is defined. Cartesian, Parametric and Polar forms need different approaches.

Common Mistakes to Avoid

Mistake 1: Forgetting to find all intersection points. This leads to incorrect limits of integration.

Mistake 2: Incorrectly identifying the "top" and "bottom" curves. Always sketch the graph to be sure!

Mistake 3: Not taking the absolute value when needed. Area is always positive!

Mistake 4: Mixing up variables of integration. If you're integrating with respect to $x$ , everything must be in terms of $x$ !

Mistake 5: Messing up the derivatives in Parametric form

JEE-Specific Tricks:

Weierstrass Substitution: For integrals involving trigonometric functions, the Weierstrass substitution ( $t = \tan(\frac{x}{2})$ ) can sometimes simplify the integral.
Leibniz Rule: If the limits of integration are also functions of a variable, use Leibniz rule for differentiating under the integral sign to simplify the problem.
Properties of Definite Integrals: Remember King and Queen properties! They can save a LOT of time! $\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b-x) dx$ and $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$

By mastering these techniques and avoiding common pitfalls, you'll be well-equipped to tackle any "Mixed Problems" question on the JEE Main. Keep practicing, and all the best!