Math topic中文

Concept limit of sum

Concept of definite integral as a limit of sum.

Practice questions

1The diagram above shows a sketch of the graph of $y=f(x)$ for $x \\ge 0$ . There are $n$ rectangles each of width $h$ dra…
2Find the exact value of $\pi \int_{1}^{6} x^3 \text{ d}x$ . The diagram shows a sketch of the curve $y = x^{\frac{3}{2}}$ …
3Find $\int \frac{x}{\sqrt{2x-1}}\,dx.$ Using the substitution $t = \tan x$ , find $\int \frac{1}{4\cos^{2}x + 9\sin^{2}x}\,dx.$ …
4The graph of $y = \sqrt{x+40}$ is shown in the diagram below. By considering the shaded rectangle, show that $\sqrt{n+40} < \int_n^{n+1} \sqrt{x+40}\, dx.$ …
5Let $f(x)=\dfrac{x}{\sqrt{x+1}}$ . Show that $\displaystyle \int_{0}^{1} f(x)\,dx=\dfrac{2}{3}\bigl(2-\sqrt{2}\bigr)$ . Th…
6The diagram above shows a sketch of the graph of $y = \text{f}(x)$ for $x \ge 0$ . There are $n$ rectangles each of width…
7The diagram above shows a sketch of the graph of $y=f(x)$ for $x\ge 0$ . There are $n$ rectangles each of width $h$ drawn…
8Section A: Pure Mathematics [40 marks] Given that f is a continuous function, explain, with the aid of a sketch, why the…