Calculate the following definite integrals:
\int_{ - 4 }^{2} e^{x} dx
\int_{0}^{3} e^{ 4 x} dx
\int_{0}^{2} e^{ \frac{3}{2} x} dx
\int_{ - 2 }^{4} e^{ 3 x} dx
\int_{ - 3 }^{5} e^{ - 2 x } dx
\int_{0}^{2} \left(7 + e^{x}\right) dx
\int_{0}^{3} \left( 2 e^{ 5 x} + e^{ - 7 x }\right) dx
\int_{ - 3 }^{3} \left(e^{ \frac{1}{4} \theta} - 3 e^{ - 0.2 \theta }\right) d\theta
\int_{3}^{4} \left(e^{ 2 x} + 4\right)^{2} dx
\int_{ - 4 }^{4} \left(e^{x} + e^{ 2 x}\right)^{2} dx
\int_{ - 4 }^{3} \dfrac{e^{ - 2 x} + e^{ 3 x}}{e^{x}} dx
\int_{1}^{16} \left(e^{ 4 x} + x^{2} - \sqrt{x}\right) dx
Calculate the following definite integrals:
\int_{\frac{3 \pi}{2}}^{ 2 \pi} \cos x \ dx
\int_{\frac{\pi}{2}}^{\pi} \left( - \sin x \right) dx
\int_{\frac{\pi}{6}}^{\frac{7 \pi}{6}} 2 \sin x \ dx
\int_{\frac{\pi}{6}}^{\frac{7 \pi}{6}} 2 \cos x \ dx
\int_{ - \frac{\pi}{6} }^{\frac{7 \pi}{6}} \sin 3 x \ dx
\int_{ - \frac{\pi}{6} }^{\frac{7 \pi}{6}} \cos 3 x \ dx
\int_{ - \frac{2 \pi}{3} }^{ 2 \pi} \sin \left(\dfrac{x}{4}\right) dx
\int_{ - \frac{4 \pi}{3} }^{\frac{2 \pi}{3}} \cos \left(\dfrac{x}{4}\right) dx
\int_{ - 6 }^{9} \sin \left(\dfrac{\pi x}{3}\right) dx
\int_{ - 8 }^{4} \cos \left(\dfrac{\pi x}{2}\right) dx
\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \sin \left(x + \pi\right) dx
\int_{ - \frac{\pi}{2} }^{\pi} \cos \left( 4 x - \dfrac{\pi}{2}\right) dx
\int_{0}^{\frac{\pi}{6}} \left(\sin x + \cos x\right) dx
\int_{ - \frac{\pi}{6} }^{\frac{\pi}{6}} \left( 4 \cos x + \cos 4 x\right) dx
\int_{ - \pi }^{\pi} \left(\sin \left(\dfrac{\theta}{6}\right) - \cos \left(\dfrac{\theta}{6}\right)\right) d\theta
\int_{0}^{\frac{\pi}{3}} \left( 2 \cos 3 x + \dfrac{1}{4} \sin 2 x\right) dx
\int_{ - \frac{\pi}{2} }^{\frac{\pi}{2}} \left( 8 \cos \left( 2 x\right) - 9 \sin \left( 3 x\right)\right) dx
Consider the function y = e^{x} + e^{ - x }.
Find y'.
Find the exact value of \int_{0}^{3} \dfrac{e^{x} - e^{ - x }}{e^{x} + e^{ - x }} dx.
Consider the function y = e^{ 3 x} \left(x - \dfrac{1}{3}\right).
Find y'.
Hence find the exact value of \int_{6}^{9} x e^{ 3 x} dx.
Calculate the exact area of the region enclosed between the following:
y = e^{ 4 x}, the x-axis, and the lines \\ x = -1 and x=2.
y = e^{x}, the coordinate axes and the line x = 2.
y = - e^{x}, the coordinate axes and the line x = 5.
y = e^{x} + 4, the coordinate axes and the line x = 4.
y = e^{ - 6 x}, the x-axis, the lines x = - 2 and x = 1.
y = 6 e^{ 3 x}, the x-axis, the lines x = - 3 and x=3.
y = e^{\frac{x}{4}} + e^{ - \frac{x}{4} }, the coordinate axes and the line x = \dfrac{1}{2}.
Calculate the exact area of the regions bounded by the x-axis and the curve:
y = 3 \sin x, from x=0 and x=\pi.
y = \cos 3 x, from x=0 to x=\dfrac{\pi}{3}.
y = \sin 2 x, from x = \dfrac{\pi}{4} to x = \dfrac{3 \pi}{4}.
y = 3 \sin x, from x=0 to x=2\pi.
y = \sin x, from x = 0 to x = \dfrac{\pi}{2}.
y = 3 \cos x, from x = 0 to x = \dfrac{\pi}{2}.
y = \cos x, from x = 0 to x = 2 \pi.
y = 3 \left(\sin x + 1\right), from x = 0 to x = 2 \pi.
y = \dfrac{1}{3} + 3 \cos x, from x = - \dfrac{\pi}{6} to x = \dfrac{\pi}{6}.
y = 2 + \dfrac{1}{4} \sin 2 x, from x = 0 to x = \dfrac{\pi}{2}.
y = 8 \cos 2 x + 4 \sin 2 x, from x = 0 to x = \dfrac{\pi}{8}.
Consider the function f \left( x \right) = e^{ 3 x} - e^{ - 3 x }.
Find the x-intercept(s) of the function.
State the limiting function as x approaches +\infty.
Determine whether the function is odd or even.
Find the exact area enclosed between the curve, the coordinate axes, and the line x = 3.
Find the exact area bound by the curve, the x-axis, and the lines x = 1 and x = - 1.
Consider the integral \int_{ - \frac{\pi}{12} }^{\frac{\pi}{12}} \sin 4 x \ dx.
Evaluate the definite integral.
State the values of x on the domain - \dfrac{\pi}{12} \leq x \leq \dfrac{\pi}{12} , for where the curve y= \sin 4x is:
Above the x-axis
Below the x-axis
Hence calculate the exact area bound by the curve y= \sin 4x, the x-axis, and the lines x = - \dfrac{\pi}{12} and x = \dfrac{\pi}{12} .
Consider the curve y = \cos \left(\dfrac{x}{4}\right).
Find the area bound by the curve and the axes, between x = 0 and x = \dfrac{\pi}{2}.
Hence find the area bound by the curve and the x-axis, between x = - 4 \pi and x = 4 \pi.
Consider the curve y = 4 \sin \left(\dfrac{\pi x}{5}\right).
State the period of the curve.
Find the exact area bounded by the curve and the x-axis, between x = \dfrac{5}{4} and x = \dfrac{5}{2}.
Consider the curve y = \sin 3 x.
Graph the function on the domain 0 \leq x \leq 2 \pi.
Find the exact area of the region bounded by the curve, the x-axis and the lines x = 0 and x= \dfrac{\pi}{4}.
Consider the curve y = 3 \sin 2 x.
Graph the function on the domain 0 \leq x \leq 2 \pi.
Find the exact area bound by the curve and the x-axis between:
x = 0 and x = \dfrac{\pi}{2}
x = \dfrac{\pi}{4} and x = \dfrac{2 \pi}{3}
Consider the function y = 2 - \cos x.
Graph the function on the domain 0 \leq x \leq 2 \pi.
Hence find the area represented by the integral \int_{0}^{\frac{\pi}{2}} \left(2 - \cos x\right) dx.