1.
The area bounded by $$f\left( x \right) = {x^2},\,0 \leqslant x \leqslant 1,\,g\left( x \right) = - x + 2,\,1 \leqslant x \leqslant 2$$ and $$x$$-axis is :
3.
Let $$I = \int_{ - a}^a {\left( {p{{\tan }^3}x + q{{\cos }^2}x + r\sin \,x} \right)dx,} $$ where $$p,\,q,\,r$$ are arbitrary constants. The numerical value of $$I$$ depends on :
$$I = p\int_{ - a}^a {{{\tan }^3}x\,dx} + q\int_{ - a}^a {{{\cos }^2}x\,dx} + r\int_{ - a}^a {\sin \,x\,dx} $$
$$ = p \times 0 + 2q\int_0^a {{{\cos }^2}x\,dx} + r \times 0,$$ because $${\tan ^3}x$$ and $$\sin \,x$$ are odd functions, and $${\cos ^2}x$$ is an even function.
4.
Let $$f\left( x \right)$$ be a given integrable function such that $$f\left( {x + k} \right) = f\left( x \right)$$ for all $$x\, \in \,R.$$ Then $$\int_a^{a + k} {f\left( x \right)dx} $$ depends for its value on :
7.
The value of $$\int_{ - 2}^2 {\frac{{{{\sin }^2}x}}{{\left[ {\frac{x}{\pi }} \right] + \frac{1}{2}}}dx,} $$ where $$\left[ x \right] = $$ the greatest integer greater than or equal to $$x,$$ is :