NCERT Class 12 · Ch 7 Exercise 7.10 · Q9
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.10 Question 9
Evaluate \(\int\limits_{0}^{2} x \sqrt{2 - x}\, dx\).
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: \(\frac{16 \sqrt{2}}{15}\)
- Work on each open interval where the original integrand is real and defined.
- Put u=2−x, so x=2−u. Integrate (2−u)√u from 0 to 2.
- An antiderivative is F(x)=\(\frac{2 \left(2 - x\right)^{\frac{5}{2}}}{5} - \frac{4 \left(2 - x\right)^{\frac{3}{2}}}{3}\).
- At the upper and lower limits, F(b)=\(0\) and F(a)=\(- \frac{16 \sqrt{2}}{15}\).
- By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(\frac{16 \sqrt{2}}{15}\).
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