## Related questions with answers

(a) integrate to find F as a function of x, and (b) demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a). $F(x)=\int_{4}^{x} t^{3 / 2} d t$

Solution

Verified### a)

$\begin{align*} F(x)=&\int_4^x t^{3/2}\ d t\\ =&\left[\frac{t^{5/2}}{5/2}\right]_4^x\tag{general power and constant rules}\\ =&\frac25\left(x^{5/2}-0^{5/2}\right)\tag{substitute by the limits}\\ =&\frac25\sqrt{x^5}\tag{simplify}\\ \end{align*}$

### b)

$\begin{align*} \dfrac{d\ }{dx}F(x)=&\dfrac{d\ }{dx}\left(\frac25\sqrt{x^5}\right)\tag{from (a)}\\ =&\frac25\dfrac{d\ }{dx}\ x^{5/2}\tag{multiple constant rule}\\ =&\frac25\cdot\frac52x^{3/2}=x^{3/2}=f(x)\tag{power rule}\\ \end{align*}$

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Consider the region bounded by the graphs of f(x) = 8x / (x+1), x=0, x=4 and y=0. (a) Redraw the figure, and complete and shade the rectangles representing the lower sum when n=4.Find this lower sum. (b) Redraw the figure, and complete and shade the rectangles representing the upper sum when n=4. Find this upper sum. (c) Redraw the figure, and complete and shade the rectangles whose heights are determined by the functional values at the midpoint of each subinterval when n=4. Find this sum using the Midpoint Rule. (d) Verify the following formulas for approximating the area of the region using n subintervals of equal width. Lower sum: s(n) = Σ_(i=1)^n f(i=1) 4/n Upper sum: S(n) = Σ_(i=1)^n f(i)4/n Midpoint Rule: M(n) = Σ_(i=1)^n f[(i-1/2) 4/n] (4/n) (e) Use a graphing utility to create a table of values of s(n), S(n), and M(n) for n = 4, 8, 20, 100, and 200. (f) Explain why s(n) increases and S(n) decreases for increasing values of n, as shown in the table in part (e).

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- calculus
Consider the region bounded by the graphs of f(x) = 8x / (x+1), x=0, x=4 and y=0. (a) Redraw the figure, and complete and shade the rectangles representing the lower sum when n=4.Find this lower sum. (b) Redraw the figure, and complete and shade the rectangles representing the upper sum when n=4. Find this upper sum. (c) Redraw the figure, and complete and shade the rectangles whose heights are determined by the functional values at the midpoint of each subinterval when n=4. Find this sum using the Midpoint Rule. (d) Verify the following formulas for approximating the area of the region using n subintervals of equal width. Lower sum: s(n) = Σ_(i=1)^n f(i=1) 4/n Upper sum: S(n) = Σ_(i=1)^n f(i)4/n Midpoint Rule: M(n) = Σ_(i=1)^n f[(i-1/2) 4/n] (4/n) (e) Use a graphing utility to create a table of values of s(n), S(n), and M(n) for n = 4, 8, 20, 100, and 200. (f) Explain why s(n) increases and S(n) decreases for increasing values of n, as shown in the table in part (e).

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