Find the area using Riemann Sums for $f(x)=1/x$ between $x=3$ and $x=5$ using a right sum with $2$ rectangles of equal width

Find the area using Riemann Sums for $$f(x)=1/x$$ between $$x=3$$ and $$x=5$$ using a right sum with $$2$$ rectangles of equal width

$$n=2, a=3, b=5$$. I keep getting 1/4 as my answer but my review says its 9/20. Please help

• How did you get $1/4$? Nov 15, 2019 at 16:19

Yes, the answer is $$\frac9{20}$$. After all:

• The first rectangle has its vertices at $$(3,0)$$, $$(4,0)$$, $$\left(4,\frac14\right)$$, and $$\left(3,\frac14\right)$$; therefore, its area is $$\frac14$$.
• The second rectangle has its vertices at $$(4,0)$$, $$(5,0)$$, $$\left(5,\frac15\right)$$, and $$\left(4,\frac15\right)$$; therefore, its area is $$\frac15$$.

And $$\frac14+\frac15=\frac9{20}$$.

The width of the rectangles are width $$\frac{5-3}{2}=1$$. The right Riemann sum would give you $$f(4)\times1+f(5)\times1=\frac14+\frac15=\boxed{\frac9{20}}.$$

Between $$x = 3$$ and $$x = 5$$ with two equal width rectangles means one rectangle goes from $$x = 3$$ to $$x = 4$$, and the other from $$x = 4$$ to $$x = 5$$.

The first rectangle, going from $$x = 3$$ to $$x = 4$$ has height $$\frac14$$, as that is the function value at the right side of this interval. As the width is $$1$$, the area is $$\frac14$$.

The second rectangle, going from $$x = 4$$ to $$x = 5$$, has height $$\frac15$$, as that's the function value at the right side of the interval. Thus its area is $$\frac15$$.

In total, the areas of these two rectangles is $$\frac14 + \frac15 = \frac{9}{20}$$.