Geometric Interpretation of Def.Int as Limit of Sum

Q. ${\int }_0^{\frac{\pi }{2}}log(tanx+cotx)dx=$

1. $\pi log2$
2. $-\pi log2$
3. $-\frac{\pi }{2}log2$
4. $\frac{\pi }{2}log2$

Q. If $f\left(x\right)$ is a continuous function defined on $[a,b]$ such that $f\left(x\right)\ge 0$ $x\in [a,b]$ then the area under the curve as the limit of a sum can be given as
$(b-a)\underset{n\to \infty }{lim}\frac{1}{n}\left[f\right(a)+f(a+h)...f(a+(n-1)h\left)\right]$
Where $h=\frac{b-a}{n}$ and $h\to 0$ as $n\to \infty$

1. True
2. False

Q. If $\left[x\right]$ denotes the greater integer less than or equal to x, then the value of ${\int }_1^5[|x-3|]$dx is

1. 1
2. 2
3. 4
4. 8

Q. Let $a,b,c$ be non – zero real numbers such that
${\int }_0^1(1+co{s}^{8}x)(a{x}^2+bx+c)dx={\int }_0^2(1+co{s}^{8}x)(a{x}^2+bx+c)dx.$
Then the quadratic equation $a{x}^2+bx+c=0$ has

1. Two imaginary roots
2. No root in (0, 2)
3. At least one root in (0, 2)
4. A double root in (0, 2)

Q. Let $I=\underset{a}{\overset{b}{\int }}(x^4-2x^2)dx$. If $I$ is minimum then the ordered pair $(a,b)$ is :

1. $(-\sqrt{2},0)$
2. $(-\sqrt{2},\sqrt{2})$
3. $(0,\sqrt{2})$
4. $(\sqrt{2},-\sqrt{2})$

Q. Let $a,b,c$ be non – zero real numbers such that
${\int }_0^1(1+co{s}^{8}x)(a{x}^2+bx+c)dx={\int }_0^2(1+co{s}^{8}x)(a{x}^2+bx+c)dx.$
Then the quadratic equation $a{x}^2+bx+c=0$ has

1. At least one root in (0, 2)
2. A double root in (0, 2)
3. No root in (0, 2)
4. Two imaginary roots

Q. The value of ${\int }_0^2(x-{\log }_{2}a)dx=2{\log }_2\left(\frac{2}{a}\right)$ for which of the following conditions?

1. $a>0$
2. $a>2$
3. $a=4$
4. $a=8$

Q. Using definite integration, find area of the triangle with vertices at A(1, 1), B(3, 3)A(1, 1), B(3, 3).

Q. Maximum value of g(x) in x $ϵ$ [0, 7] is.

1. 6
2. 9/2
3. 3/2
4. 3

Q. Solve ${\int }_0^{100}{e}^{x-\left[x\right]}dx=?$ where $\left[x\right]$ is greatest integer function.

1. $100e$
2. $100(e-1)$
3. $100(e+1)$
4. $100(1-e)$