let ( i=int_{0}^{1} f(x) d x ), where ( f ) is the function whose graph is shown.\n(a) use the graph and…

let ( i=int_{0}^{1} f(x) d x ), where ( f ) is the function whose graph is shown.\n(a) use the graph and trapezoid rule with ( n = 5 ) to approximate ( i=int_{0}^{1} f(x) d x ). (round your answer to two decimal places at the end.)\n(b) decide whether your answer in part (a) will underestimate or overestimate ( i ). explain your reason clearly.\nshow all steps clearly.

let ( i=int_{0}^{1} f(x) d x ), where ( f ) is the function whose graph is shown.\n(a) use the graph and trapezoid rule with ( n = 5 ) to approximate ( i=int_{0}^{1} f(x) d x ). (round your answer to two decimal places at the end.)\n(b) decide whether your answer in part (a) will underestimate or overestimate ( i ). explain your reason clearly.\nshow all steps clearly.

Answer

(a)

Step1: Calculate (\Delta x)

The formula for (\Delta x=\frac{b - a}{n}), where (a = 0), (b = 1), (n=5). (\Delta x=\frac{1-0}{5}=0.2)

Step2: Find the (x) - values

(x_0=a = 0), (x_1=x_0+\Delta x=0.2), (x_2=x_1+\Delta x = 0.4), (x_3=x_2+\Delta x=0.6), (x_4=x_3+\Delta x = 0.8), (x_5=x_4+\Delta x=1)

Step3: Estimate (f(x_i)) from the graph

From the graph (assuming we can read the (y) - values): (f(x_0)=f(0) = 0) (f(x_1)=f(0.2)\approx1.3) (f(x_2)=f(0.4)\approx1.9) (f(x_3)=f(0.6)\approx1.7) (f(x_4)=f(0.8)\approx1.1) (f(x_5)=f(1)=0)

Step4: Apply the Trapezoid Rule

The Trapezoid Rule formula is (T_n=\frac{\Delta x}{2}[f(x_0)+2f(x_1)+2f(x_2)+2f(x_3)+2f(x_4)+f(x_5)]) Substitute (\Delta x = 0.2) and the (f(x_i)) values: [ \begin{align*} T_5&=\frac{0.2}{2}[0 + 2\times1.3+2\times1.9+2\times1.7+2\times1.1+0]\ &=0.1[0 + 2.6+3.8+3.4+2.2]\ &=0.1\times12\ &=1.20 \end{align*} ]

(b)

Brief Explanations:

The function (y = f(x)) is concave - down (the graph is a parabola opening downwards). The Trapezoid Rule over - estimates the integral of a concave - down function. The trapezoids used in the Trapezoid Rule for a concave - down function lie above the curve.

Answer:

(a) (1.20) (b) The answer in part (a) overestimates (I). Since the function (y = f(x)) is concave - down, the trapezoids used in the Trapezoid Rule lie above the curve.