approximate the area under the curve $y = x^{3}$ from $x = 3$ to $x = 5$ using a right endpoint…
approximate the area under the curve $y = x^{3}$ from $x = 3$ to $x = 5$ using a right endpoint approximation with 4 subdivisions.
Answer
Explanation:
Step1: Calculate width of sub - intervals
The interval is from $a = 3$ to $b = 5$ and $n=4$. The width $\Delta x=\frac{b - a}{n}=\frac{5 - 3}{4}=\frac{2}{4}=0.5$.
Step2: Find right - endpoints
The right - endpoints $x_i$ of the sub - intervals are: $x_1=3 + 0.5=3.5$, $x_2=3.5 + 0.5 = 4$, $x_3=4+0.5 = 4.5$, $x_4=4.5 + 0.5=5$.
Step3: Evaluate the function at right - endpoints
$y(x_1)=(3.5)^3=42.875$, $y(x_2)=(4)^3 = 64$, $y(x_3)=(4.5)^3=91.125$, $y(x_4)=(5)^3=125$.
Step4: Calculate the sum for right - endpoint approximation
$R_4=\sum_{i = 1}^{4}y(x_i)\Delta x=\Delta x(y(x_1)+y(x_2)+y(x_3)+y(x_4))=0.5(42.875 + 64+91.125+125)=0.5\times323=161.5$.
Answer:
$161.5$