use the following table to estimate ∫₀¹⁰ f(x)dx. assume that f(x) is a decreasing function.\n\nx 0 2 4 6 8…

use the following table to estimate ∫₀¹⁰ f(x)dx. assume that f(x) is a decreasing function.\n\nx 0 2 4 6 8 10\nf(x) 51 46 42 36 25 7\n\nto estimate the value of the integral we use the left - hand sum approximation with δx =\n\nthen the left - hand sum approximation is\n\nto estimate the value of the integral we can also use the right - hand sum approximation with δx =\n\nthen the right - hand sum approximation is\n\nthe of the left - and right sum approximations is a better estimate which is

use the following table to estimate ∫₀¹⁰ f(x)dx. assume that f(x) is a decreasing function.\n\nx 0 2 4 6 8 10\nf(x) 51 46 42 36 25 7\n\nto estimate the value of the integral we use the left - hand sum approximation with δx =\n\nthen the left - hand sum approximation is\n\nto estimate the value of the integral we can also use the right - hand sum approximation with δx =\n\nthen the right - hand sum approximation is\n\nthe of the left - and right sum approximations is a better estimate which is

Answer

Explanation:

Step1: Determine $\Delta x$

The $x$-values in the table increase by 2. So, $\Delta x=2$.

Step2: Calculate left - hand sum

The left - hand sum $L$ for $\int_{0}^{10}f(x)dx$ with $n = 5$ and $\Delta x=2$ is given by $L=\Delta x\sum_{i = 0}^{4}f(x_i)$. [ \begin{align*} L&=2\times(f(0)+f(2)+f(4)+f(6)+f(8))\ &=2\times(51 + 46+42+36+25)\ &=2\times200\ &=400 \end{align*} ]

Step3: Calculate right - hand sum

The right - hand sum $R$ for $\int_{0}^{10}f(x)dx$ with $n = 5$ and $\Delta x=2$ is given by $R=\Delta x\sum_{i = 1}^{5}f(x_i)$. [ \begin{align*} R&=2\times(f(2)+f(4)+f(6)+f(8)+f(10))\ &=2\times(46 + 42+36+25+7)\ &=2\times156\ &=312 \end{align*} ]

Step4: Find a better estimate

The average of the left - hand and right - hand sums is a better estimate. The average $A=\frac{L + R}{2}=\frac{400+312}{2}=\frac{712}{2}=356$.

Answer:

To estimate the value of the integral we use the left - hand sum approximation with $\Delta x = 2$. Then the left - hand sum approximation is $400$. To estimate the value of the integral we can also use the right - hand sum approximation with $\Delta x = 2$. Then the right - hand sum approximation is $312$. The average of the left - and right sum approximations is a better estimate which is $356$.