Which of the following integrals represents the length of the curve $$y = \sqrt{x}$$ on the interval $$[1, 4]$$?
Area between curves, volumes of revolution, average value, and arc length.
12 questions
Which of the following integrals represents the length of the curve $$y = \sqrt{x}$$ on the interval $$[1, 4]$$?
$$\displaystyle\int_{1}^{4} \sqrt{1 - \frac{1}{4x}} dx$$
$$\displaystyle\int_{1}^{4} \sqrt{1 + \frac{1}{4}x^2} dx$$
$$\displaystyle\int_{1}^{4} \sqrt{1 + \frac{1}{2\sqrt{x}}} dx$$
$$\displaystyle\int_{1}^{4} \sqrt{1 + \frac{1}{4x}} dx$$
Which of the following is an example of correctly approximating the area under the curve $$f(x)=\sqrt{x}$$ from $$x=1$$ to $$x=4$$ using a definite integral?
Area = $$4^{3/2}-1^{3/2}$$
Area = $$\frac{2}{3}(4-1)=2$$
Area = $$\frac{2}{3}(4^{3/2}-1^{3/2})=\frac{14}{3}$$
Area = $$\frac{1}{2}(4^{3/2}-1^{3/2})$$
Let R be the region in the first quadrant bounded by the graph of y = sin(x), the x-axis, and the line x = π/2. If the area of R is given by $\displaystyle\int_{0}$^(π/2) sin(x)dx = 1, what is the area of the region formed by reflecting R across the y-axis?
2
π/2
1
0
Let f and g be continuous functions such that f(x) ≥ g(x) on the interval [0, a]. If $\displaystyle\int_{0}$^a f(x)dx = 12 and $\displaystyle\int_{0}$^a g(x)dx = 5, what is the area of the region bounded by the graphs of f and g between x = 0 and x = a?
17
7
60
12 - 5a
What is the area of the region enclosed by the graph of y = x², the x-axis, and the vertical lines x = 0 and x = 2?
8.33
1.5
2.67
4.16
The accumulation of change of a linear function over an interval can be evaluated by finding the area of which geometric shape?
Triangle
Rectangle
Circle
Parabola
The differentiable function $$h$$ models Tara's heart rate in beats per minute, where $$t$$ is the time in minutes since the start of a workout. Which of the following expressions gives Tara's average heart rate on the interval from $$t=30$$ to $$t=60$$?
$$\frac{1}{30} \int_{30}^{60} h(t) dt$$
$$\frac{h'(30) + h'(60)}{2}$$
$$\int_{30}^{60} h(t) dt$$
$$\frac{1}{30} \int_{30}^{60} h'(t) dt$$
The derivative of a function $$f$$ is given by $$f'(x) = x^3 - 6x^2 + 9x - 2$$. Which expression represents the area of the region enclosed by the graph of $$f'(x)$$ and the x-axis on the interval $$[0,3]$$?
|f(3) - f(0)|
$$\int_{0}^{3} |f'(x)| dx$$
f(3) - f(0)
f(3) + f(0)
The area under the curve of a non-negative function represents which of the following?
The average rate of change
The maximum rate of change
The instantaneous rate of change
The accumulation of change
The region enclosed by the graphs of $$y = x^2$$ and $$y = 4$$ is revolved about the y-axis. Using the shell method, which of the following integrals gives the volume of the resulting solid?
$\displaystyle\int_{0}^{4}$ π(2√y)² dy
π$\displaystyle\int_{0}^{4}$ (2√y) dy
$\displaystyle\int_{0}^{2}$ π[(4 - x²)² - x⁴] dx
$\displaystyle\int_{0}^{2}$ 2πx(4 - x²) dx
What is the volume of the solid generated when the region enclosed by the graphs of $$y = x^2$$ and $$y = 4$$ is rotated about the y-axis?
32π/3
16π/3
4π
8π
The region bounded by the graph of $$y = \sqrt{x}$$, the x-axis, and the line $$x = 4$$ is revolved about the x-axis. Which of the following expressions represents the volume of the resulting solid as determined by the disk method?
$$\int_{0}^{2} 2\pi y dy$$
$$\int_{0}^{4} 2\pi x dx$$
$$\int_{0}^{2} \pi y^2 dy$$
$$\int_{0}^{4} \pi (\sqrt{x})^2 dx = \int_{0}^{4} \pi x dx$$