Q. 45 - Calculus

Question

Suppose that, as in Section 4.1, you drive in a car for 40 seconds with velocity $v\left(t\right)=-0.22t^2+88t$ feet per second second, as shown in the graph that follows. If your total

distance travelled is equal to the area under the velocity curve on [0, 40], then find lower and upper bounds for your distance travelled by using

(a) the lower sum with n = 4 rectangles;

(b) the upper sum with n = 4 rectangles.

Step-by-Step Solution

Answer

(a) The lower sum with $n=4$ rectangles is $464.4$

(b) The upper sum with $n=4$ rectangles is $2811.07$

Step 1. Given Information

The velocity of the car is  and total distance travelled is equal to the area under the velocity curve on [0, 40],

Step 2. Finding ∆ x

$∆x=\frac{b-a}{n}=\frac{40-0}{4}=10$

$x_k=a+k∆x=0+10k=10k$

Part (a) Step 1. Finding Lower Sum

$$\begin{gathered} f\left(0\right)∆x+f\left(10\right)∆x+f\left(20\right)∆x+f\left(30\right)∆x \\ =0\times 10+366.67+1173.33-1075.6=464.4 \end{gathered}$$

Part (b) Step 1. Finding Upper Sum

$$\begin{gathered} f\left(10\right)∆x+f\left(20\right)∆x+f\left(30\right)∆x+f\left(40\right)∆x \\ =366.67+1173.33-1075.6+2346.67=2811.07 \end{gathered}$$