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Q1(i):
Find the area under the given curves and given lines:
(i) $y = x^2$, $x = 1$, $x = 2$ and $x$-axis
Solution :
Step 1: Identifying the Region and Theoretical Setup
The problem asks for the area of the region bounded by the parabola $y = x^2$, the vertical lines $x = 1$ and $x = 2$, and the $x$-axis. Since the function $y = x^2$ is continuous and non-negative for all $x \in [1, 2]$, the area $A$ is given by the definite integral of the function $f(x)$ with respect to $x$ over the interval $[a, b]$.
Formula: $A = \int_{a}^{b} |f(x)| \, dx = \int_{1}^{2} x^2 \, dx$ [Per the Fundamental Theorem of Calculus].
Step 2: Performing the Integration
Apply the Power Rule for integration, $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, where $n = 2$.
$$A = \int_{1}^{2} x^2 \, dx$$
$$A = \left[ \frac{x^{2+1}}{2+1} \right]_{1}^{2}$$
$$A = \left[ \frac{x^3}{3} \right]_{1}^{2}$$
Step 3: Applying Limits of Integration
Per the Second Fundamental Theorem of Calculus, substitute the upper limit and subtract the evaluation at the lower limit:
$$A = \left( \frac{2^3}{3} \right) - \left( \frac{1^3}{3} \right)$$
$$A = \frac{8}{3} - \frac{1}{3}$$
$$A = \frac{7}{3}$$
Summary of Calculation
| Component | Value |
|---|---|
| Function $f(x)$ | $x^2$ |
| Lower Bound ($a$) | $1$ |
| Upper Bound ($b$) | $2$ |
| Resulting Area | $7/3$ square units |
Final Solution: The area under the curve is $\frac{7}{3}$ square units.
More Questions from Class 12 Mathematics Application of Integrals Miscellaneous Exercise on Chapter 8
- Q1(ii): Find the area under the given curves and given lines: (ii) $y = x^4$, $x = 1$, $x = 5$ and $x$-axis
- Q2: Sketch the graph of $y = |x + 3|$ and evaluate $\int_{-6}^{0} |x + 3| dx$.
- Q3: Find the area bounded by the curve $y = \sin x$ between $x = 0$ and $x = 2\pi$.
- Q4: Area bounded by the curve $y = x^3$, the $x$-axis and the ordinates $x = – 2$ and $x = 1$ is
- Q5: The area bounded by the curve $y = x | x |$, $x$-axis and the ordinates $x = – 1$ and $x = 1$ is given by [Hint : $y = x^2$ if $x > 0$ and $y = – x^2$ if $x < 0$].
CBSE Solutions for Class 12 Mathematics Application of Integrals
Chapters in CBSE - Class 12 Mathematics
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