Mathematics Pure Math 2,3 9709: INTEGRATION Topical Past Paper Questions

(a) Use the substitution $ x = 2 \sin u $ to show that $$ \int_{0}^{1}\frac{3x^{3}}{\sqrt{4-x^{2}}}\mathrm{d}x=\int_{a}^{b}24\sin^{3}u\mathrm{d}u, $$ where $a$ and $b$ are exact values to be found. (b) Hence, find the exact value of $ \int_{0}^{1}\frac{3x^{3}}{\sqrt{4-x^{2}}}dx $…

The constant $a$ is such that $\int_{0}^{a} x e^{\frac{1}{2}x} \, dx = 6$. (a) Show that $a = 2 + e^{-\frac{1}{2}a}$. [5] (b) Verify by calculation that $a$ lies between 2.2 and 2.4. [2] (c) Use an iterative formula based on the equation in part (a) to determine $a$ correct to 2…

The diagram shows the curve $ y = 2\sin x\sqrt{2 + \cos x} $, for $ 0 \leq x \leq 2\pi $, and its minimum point M, where x = a. (a) Find the value of $a$ correct to 2 decimal places. 10^{6} (b) Use the substitution $u=2+\cos x$ to find the exact area of the shaded region $R$. [6]…