Home > PC3 > Chapter 13 > Lesson 13.2.2 > Problem13-73

13-73.

Given $f\left(x\right) = −x^{2} + 12x$, complete the following problems.

1. Approximate $A(f,0\le x\le6)$ using $16$ left endpoint rectangles.

sub-interval width $=\frac{6-0}{16}=\frac{3}{8}$

$x_1=0,\ x_2=\frac{3}{8},\ x_3=\frac{6}{8},\ ...\ x_n=?$

$\text{area }=\displaystyle\sum_{n=0}^{15}\frac{3}{8} f(x_n)$

2. Approximate $A(f,1.2\le x\le12)$ using $21$ right endpoint rectangles.

sub-interval width $=\frac{12-1.2}{21}=w$

$x_1=1.2,\ x_2=1.2+w,\ x_3=1.2+2w,\ ...\ x_n=?$

$\text{area }=\displaystyle\sum_{n=1}^{21}w\cdot f(x_n)$

3. Approximate $A(f,0\le x\le10)$ using $15$ trapezoids.

Average the areas of the left and right endpoint rectangle approximations.