91Ó°ÊÓ

Find the area of the regions. Between \(y=x^{2}\) and \(y=x^{3}\) for \(0 \leq x \leq 1\).

(a) Differentiate \(x^{3}+x\) (b) Use the Fundamental Theorem of Calculus to find \(\int_{0}^{2}\left(3 x^{2}+1\right) d x\)

At time, \(t,\) in seconds, your velocity, \(v,\) in meters/second, is given by $$ v(t)=1+t^{2} \quad \text { for } \quad 0 \leq t \leq 6 $$ Use \(\Delta t=2\) to estimate the distance traveled during this time. Find the upper and lower estimates, and then average the two.

For time, \(t,\) in hours, \(0 \leq t \leq 1,\) a bug is crawling at a velocity, \(v,\) in meters/hour given by $$ v=\frac{1}{1+t} $$ Use \(\Delta t=0.2\) to estimate the distance that the bug crawls during this hour. Find an overestimate and an underestimate. Then average the two to get a new estimate.

Use a calculator or a computer to find the value of the definite integral. $$\int_{0}^{3} \ln \left(y^{2}+1\right) d y$$

(a) What is the derivative of \(\sin t ?\) (b) The velocity of a particle at time \(t\) is \(v(t)=\cos t\) Use the Fundamental Theorem of Calculus to find the total distance traveled by the particle between \(t=0\) and \(t=\pi / 2\).

Find the area of the regions. Between \(y=x^{1 / 2}\) and \(y=x^{1 / 3}\) for \(0 \leq x \leq 1\)

Use a calculator or a computer to find the value of the definite integral. $$\int_{0}^{1} \sin \left(t^{2}\right) d t$$

Find the area of the regions. Between \(y=\sin x+2\) and \(y=0.5\) for \(6 \leq x \leq 10\)

(a) If \(F(t)=\frac{1}{2} \sin ^{2} t,\) find \(F^{\prime}(t)\) (b) Find \(\int_{0.2}^{0.4} \sin t \cos t d t\) two ways: (i) Numerically. (ii) Using the Fundamental Theorem of Calculus.