Question

Evaluate: ${\int }_a^{b}xdx$ using limit of sum.

• A. $\frac{b^2-a^2}{3}$
• B. $\frac{b^2+a^2}{2}$
• C. $\frac{b^2-a^2}{2}$
• D. None of these

Answer 1

Answer: (C)

$\frac{b^2-a^2}{2}$

$I={\int }_a^{b}xdx$

$=\underset{x\to \infty }{lim}\frac{b-a}{n}(a+(a+h)+(a+2h)+a(n-1\left)h\right)$

$=(b-a)\underset{n\to \infty }{lim}\frac{1}{n}(na+(h\left)\right(1+2+......(n-e)\left)\right)$

Since $h=\frac{b-a}{n}$

$=(b-a)\underset{n\to \infty }{lim}\frac{1}{n}(na+\frac{(b-a)}{n})\frac{n(n-1)}{2}$

$(b-a)\underset{x\to \infty }{lim}\left(a\right(b-a\left)\frac{(1-\frac{1}{x})}{2}\right)$

$=(b-a(a+\frac{b-a}{2})$

$=\frac{(b-a)(b+a)}{2}$

$=\frac{b^2-a^2}{2}$ Answer