Calculus II

Integration

Indefinite Integrals

Example: Determine the integral

(1)
\int te^{\sqrt{t}}\ dt

Walpha Query: integral t*e^(sqrt(t))

Shows: The integral (with a Show steps option), plots of the indefinite integral function, alternate forms of the function, and a series expansion at t=0 (with a More terms option).

Other Examples: Not every integral is computed the way they would generally be done by hand. The steps in the following examples are not the usual technique of integration:

(2)
\sqrt{1+\left(\frac{2}{3\sqrt[3]{x}}\right)^2}
(3)
\sin^4(x)\cos^4(x)

Definite Integral

Example: Compute the integral

(4)
\int_0^1 xe^t\ dt

Walpha Query: integral from t=0 to t=1 of x*e^t

Parametric Curves

Plotting

Example: Plot the parametric curve

(5)
x(t)=t^2-1,\quad y(t)=(t-1)^2+2

from t=-2 to t=3.

Walpha Query:

plot [t^2-1,(t-1)^2+2] from t=-2 to t=3

Polar Curves

Plotting

Example: Plot the polar curve

(6)
r(\theta)=5+\sin(8\theta),\quad \theta\in [0,2\pi]

Walpha Query: polar plot r=5+sin(8t) from t=0 to t=2*pi

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