1) Sketch the region enclosed by \(\displaystyle 2y = 4 \sqrt{x}, \;\ y=5\), and \(\displaystyle 2y + 3 x = 7\)
Decide whether to integrate with respect to x or y, and then find the area of the region.
The area is?
2) Consider the area between the graphs \(\displaystyle x+ 2 y = 18\) and \(\displaystyle x + 6 = y^{2}\). This area can be computed in two different ways using integrals
\(\displaystyle \int_{a}^{b} f(x)dx + \int_{b}^{c} g(x)dx\)
Alternatively this area can be computed as a single integral
\(\displaystyle \int_{\alpha}^{\beta} h(y)dy\)
where \(\displaystyle \alpha\) = ? \(\displaystyle \beta =\)?
h(y) =?
3) Find the volume of the solid formed by rotating the region enclosed by
\(\displaystyle x=0, \quad x=1, \quad y=0, \quad y= 5 +x^{8}\)
about the x-axis.
Volume?
4)The volume of the solid obtained by rotating the region enclosed by
\(\displaystyle y = \frac{1}{x^{3}}, \quad y = 0, \quad x = 2, \quad \mbox{ and } x = 7\),
about the line y=-3 can be computed using the method of disks or washers via an integral
\(\displaystyle \displaystyle V = \int_{a}^{b}\) ?
with limits of integration a = ? and b = .?
The volume is V =? cubic units
5) Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.
\(\displaystyle x+y=4, \quad x = 5-(y-1)^{2}\) ;
about the x-axis.
Volume = ?
6) A car drives down a road in such a way that its velocity ( in m/s) at time t (seconds) is
\(\displaystyle v(t) = t^{1/2} + 4\)
.
Find the car's average velocity (in m/s) between t = 3 and t = 8.
7) One fine day in New York the low temperature occurs at 5 a.m.
and the high temperature at 5 p.m. The temperature varies
sinusoidally all day.
The temperature t hours after midnight is
\(\displaystyle T(t) = A + B \sin \left( \frac{\pi (t-C)}{12} \right)\)
where A, B, and C are certain constants.
The low temperature is 40 and the high temperature is 60 (Fahrenheit).
Find the average temperature during the first 6 hours after noon.
Hint: The high and low temperatures can be used together to find
A and B. Determine C from the fact that it is hottest at 5 p.m.
Decide whether to integrate with respect to x or y, and then find the area of the region.
The area is?
2) Consider the area between the graphs \(\displaystyle x+ 2 y = 18\) and \(\displaystyle x + 6 = y^{2}\). This area can be computed in two different ways using integrals
\(\displaystyle \int_{a}^{b} f(x)dx + \int_{b}^{c} g(x)dx\)
Alternatively this area can be computed as a single integral
\(\displaystyle \int_{\alpha}^{\beta} h(y)dy\)
where \(\displaystyle \alpha\) = ? \(\displaystyle \beta =\)?
h(y) =?
3) Find the volume of the solid formed by rotating the region enclosed by
\(\displaystyle x=0, \quad x=1, \quad y=0, \quad y= 5 +x^{8}\)
about the x-axis.
Volume?
4)The volume of the solid obtained by rotating the region enclosed by
\(\displaystyle y = \frac{1}{x^{3}}, \quad y = 0, \quad x = 2, \quad \mbox{ and } x = 7\),
about the line y=-3 can be computed using the method of disks or washers via an integral
\(\displaystyle \displaystyle V = \int_{a}^{b}\) ?
with limits of integration a = ? and b = .?
The volume is V =? cubic units
5) Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.
\(\displaystyle x+y=4, \quad x = 5-(y-1)^{2}\) ;
about the x-axis.
Volume = ?
6) A car drives down a road in such a way that its velocity ( in m/s) at time t (seconds) is
\(\displaystyle v(t) = t^{1/2} + 4\)
.
Find the car's average velocity (in m/s) between t = 3 and t = 8.
7) One fine day in New York the low temperature occurs at 5 a.m.
and the high temperature at 5 p.m. The temperature varies
sinusoidally all day.
The temperature t hours after midnight is
\(\displaystyle T(t) = A + B \sin \left( \frac{\pi (t-C)}{12} \right)\)
where A, B, and C are certain constants.
The low temperature is 40 and the high temperature is 60 (Fahrenheit).
Find the average temperature during the first 6 hours after noon.
Hint: The high and low temperatures can be used together to find
A and B. Determine C from the fact that it is hottest at 5 p.m.
