Preparation with RD Sharma Class 12 Solutions Chapter 21
FAQs on RD Sharma Class 12 Solutions Chapter 21 - Areas of Bounded Regions (Ex 21.4) Exercise 21.4
1. What is the first and most crucial step when solving problems from RD Sharma Class 12 Solutions for Exercise 21.4?
The first and most crucial step is to draw a rough sketch of the curves provided in the question. Visualising the bounded region helps in correctly identifying the points of intersection, the limits of integration, and which function represents the upper boundary versus the lower boundary.
2. How do you set up the definite integral for finding the area between a curve y = f(x) and the x-axis from x=a to x=b?
To find the area between a curve y = f(x) and the x-axis from x=a to x=b, you set up the definite integral as Area = ∫ₐᵇ y dx or Area = ∫ₐᵇ f(x) dx. It's important to take the absolute value of the result if the region lies below the x-axis, as area must be a positive quantity.
3. What is the correct method for finding the area bounded by two curves, like a parabola and a line, as shown in Chapter 21?
The step-by-step method to find the area between two curves, y = f(x) and y = g(x), is as follows:
- Find intersection points: Solve the two equations simultaneously to find the x-coordinates of their intersection, say 'a' and 'b'. These will be your limits of integration.
- Identify upper and lower curves: In the interval [a, b], determine which function has greater y-values (the upper curve) and which has lesser y-values (the lower curve).
- Set up the integral: The area is calculated using the formula: Area = ∫ₐᵇ [f(x) - g(x)] dx, where f(x) is the upper curve and g(x) is the lower curve.
4. Why is sketching the graph so essential before calculating the area of a bounded region?
Sketching the graph is essential because it prevents common errors and clarifies the problem. A sketch helps you to:
- Correctly identify the exact enclosed region whose area needs to be calculated.
- Visually confirm the points of intersection, which define the limits of integration.
- Clearly distinguish the upper boundary curve from the lower boundary curve over the entire interval, which is critical for setting up the integral `∫(upper curve - lower curve)dx` accurately.
5. How do you decide whether to integrate with respect to x (using vertical strips) or with respect to y (using horizontal strips)?
The choice depends on which method simplifies the calculation:
- Use vertical strips (dx) when the region's upper and lower boundaries are consistently defined by functions of x (i.e., y = f(x) and y = g(x)). This is the most common approach.
- Use horizontal strips (dy) when the region's right and left boundaries are more easily defined by functions of y (i.e., x = f(y) and x = g(y)). This is particularly useful for parabolas that open sideways, like y² = 4ax.
6. What is a common mistake students make when finding the area between two intersecting curves using the solutions for Exercise 21.4?
A very common mistake is incorrectly identifying the upper and lower curves. If the curves intersect and switch positions (i.e., the upper curve becomes the lower curve), the integral must be split into multiple parts at the point of intersection. Forgetting to do this and using a single integral `∫(f(x) - g(x))dx` over the entire range will lead to an incorrect answer.
7. Do the solutions for RD Sharma Exercise 21.4 cover finding the area of regions bounded by standard shapes like circles and ellipses?
Yes, the solutions for RD Sharma Class 12 Chapter 21, including Exercise 21.4, provide solved examples for finding the area of regions bounded by standard curves. This includes problems involving circles, parabolas, and ellipses, often in combination with lines, which aligns with the CBSE 2025-26 syllabus for Applications of Integrals.
