Question

The approximate value of ${\int }_{-3}^3{x}^{2}dx$ using Trapezoidal rule and taking $6$ equal intervals is?

• A. $11.5$
• B. $19$
• C. $115$
• D. $120$

Answer 1

The correct option is B $19$

${\int }_{-3}^3{x}^{2}dx$ Trapizoidal Rule

$\Rightarrow {\int }_a^{b}f\left(x\right)dx=\frac{\Delta x}{2}\left(f\right(x_0)+2f(x_1)+2f(x_2)+........f(x_6\left)\right)$

6 equal intervals

$x_1-2⟶-1x_2-1⟶0x_{3}0⟶1x_{4}1⟶2x_{5}2⟶3\Rightarrow \Delta x=\lfloor \frac{b-a}{n}\rfloor =\frac{3-(-3)}{6}=\frac{6}{6}=1x_0=-3x_1=-2x_2=-1x_3=0x_4=1x_5=2x_6=3f\left(x\right)=x^3\Rightarrow {\int }_{-3}^3{x}^{2}dx=\frac{1}{2}\Rightarrow (9+8+2+0+2+8+9)=\frac{38}{2}=19\Rightarrow {\int }_{-3}^3{x}^{2}dx=19$