Find $\displaystyle\int_{10}^{12} \ln t d t$ two ways:
(i) Numerically. (ii) Using the Fundamental Theorem of Calculus.

Find the area of the regions between the curve and the horizontal axis.

Under the curve $y = \cos\sqrt{x}$ for $0 \leq x \leq 2.$

Let $f(t)=F'(t).$ Write the integral $\int_{a}^{b}f(t)\ dt$ and evaluate it using the Fundamental Theorem of Calculus.

$$F(t) = t^{4}, a = −1, b = 1$$

Are the statements true for all continuous functions $f(x)$ and $g(x)$ ? Give an explanation for your answer.
If $f(x) \leq g(x)$ on the interval $[a, b]$, then the average value of $f$ is less than or equal to the average value of $g$ on the interval $[a, b]$.

As coal deposits are depleted, it becomes necessary to strip-mine larger areas for each ton of coal. The figure shows the number of acres of land per million tons of coal that will be defaced during strip-mining as a function of the number of million tons removed, starting from the present day.

Use your answers to the previous exercises to get a better estimate of the actual number of acres defaced.

As coal deposits are depleted, it becomes necessary to strip-mine larger areas for each ton of coal. The figure shows the number of acres of land per million tons of coal that will be defaced during strip-mining as a function of the number of million tons removed, starting from the present day.

Re-estimate the number of acres defaced using rectangles above the curve.

As coal deposits are depleted, it becomes necessary to strip-mine larger areas for each ton of coal. The figure shows the number of acres of land per million tons of coal that will be defaced during strip-mining as a function of the number of million tons removed, starting from the present day.

Estimate the total number of acres defaced in extracting the next 4 million tons of coal (measured from the present day). Draw four rectangles under the curve, and compute their area.

Let $f(t)=F'(t).$ Write the integral $\int_{a}^{b}f(t) \ dt$ and evaluate it using the Fundamental Theorem of Calculus.

$$F(t) = \tan t; a = 0, b = \pi$$

Is the statement true for all continuous functions $f(x)$ and $g(x)?$ Explain your answer.

If $\int_{2}^{6}f(x) \ dx \leq \int_{2}^{6}g(x) \ dx,$ then $f(x) \leq g(x)$ for $2 \leq x \leq 6.$

Your velocity is $v(t)=\sin \left(t^2\right) \mathrm{mph}$ for $0 \leq t \leq 1.1$. Estimate the distance traveled during this time.

Find the area of the regions between the curve and the horizontal axis.

Under $y = \ln x$ for $1 \leq x \leq 4.$

For the following problem, find the difference between the upper and lower estimates of the distance traveled at velocity $f(t)$ on the interval $a \leq t \leq b$ for $n$ subdivisions.

$$f(t) = 5t + 8, a = 1, b = 3, n = 100$$

Roger runs a marathon. His friend Jeff rides behind him on a bicycle and clocks his speed every $15$ minutes. Roger starts out strong, but after an hour and a half he is so exhausted that he has to stop. Jeff’s data follow:

(c ) How often would Jeff have needed to measure Roger’s speed in order to find lower and upper estimates within $0.1$ mile of the actual distance he ran?

The rate at which the world’s oil is consumed (in billions of barrels per year) is given by $r = f(t),$ where $t$ is in years and $t = 0$ is the start of $2004.$

(c ) Interpret each of the five terms in the sum from part (b) in terms of oil consumption.