Integrals & the power rule
- Find the indefinite integral: \(\int 7\,dx\)
\[ \int 7\,dx = 7x+C \]
- Find the indefinite integral: \(\int 5x^3\,dx\)
\[ \int 5x^3\,dx = \frac{5}{4}x^4+C \]
- Find the indefinite integral: \(\int(2x^2+6x)\,dx\)
\[ \int(2x^2+6x)\,dx = \frac{2}{3}x^3+3x^2+C \]
Initial value problems
- Given \(\frac{dy}{dx}=6x-1\), find \(y(x)\) if \(y(2)=5\).
\[ y=3x^2-x+C \] \[ y(2)=3(2)^2-2+C=10+C=5 \Rightarrow C=-5 \] \[ y=3x^2-x-5 \]
- Given \(\frac{dy}{dx}=\frac{8}{x^3}\), find \(y(x)\) if \(y(2)=1\).
Rewrite \(\frac{8}{x^3}=8x^{-3}\): \[ y=\frac{8x^{-2}}{-2}+C=-4x^{-2}+C=-\frac{4}{x^2}+C \] \[ y(2)=-1+C=1 \Rightarrow C=2 \] \[ y=-\frac{4}{x^2}+2 \]
Definite integrals
- Evaluate: \(\int_0^3 2x\,dx\)
\[ \int_0^3 2x\,dx = x^2\Big|_0^3 = 9-0 = 9 \]
- Evaluate: \(\int_1^3 (x^2-1)\,dx\)
\[ \int_1^3 (x^2-1)\,dx = \left[\frac{1}{3}x^3-x\right]_1^3 = \left(9-3\right)-\left(\frac{1}{3}-1\right) = 6-\left(-\frac{2}{3}\right) = \frac{20}{3} \]
Area between curves
- The area between \(y=4-x^2\) and \(y=x\) on \([0,1]\) is given by \(\int_0^1[(4-x^2)-x]\,dx\). Evaluate this integral.
\[ \int_0^1(4-x^2-x)\,dx = \left[4x-\frac{1}{3}x^3-\frac{1}{2}x^2\right]_0^1 = 4-\frac{1}{3}-\frac{1}{2} = \frac{19}{6} \]
- A hiker’s velocity along a trail is \(v(t)=3t^2-2t+1\) km/hr for the first \(t\) hours. Find the total distance traveled in the first 3 hours, \(\int_0^3 v(t)\,dt\).
\[ \int_0^3(3t^2-2t+1)\,dt = \left[t^3-t^2+t\right]_0^3 = 27-9+3 = 21\text{ km} \]
Differential equations
- Classify the following differential equation by order and state whether it is an ODE or PDE: \(\frac{dy}{dx}+3y=x^2\)
This is a first-order ODE — it contains only an ordinary derivative, and the highest derivative present is first order.
- Classify: \(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0\)
This is a second-order PDE — it contains partial derivatives, and the highest derivative present is second order.
- Classify: \[ \frac{\partial^2 u}{\partial x^2}+\frac{\partial^3 u}{\partial y^3}=x \]
This is a third-order PDE — it contains partial derivatives, and even though a second-order term is also present, the order of the equation is determined by the highest derivative present, which is third order.
- Verify that \(y=e^{2x}\) is a solution to the differential equation \(y'=2y\).
\[ y'=2e^{2x}=2y \] So \(y=e^{2x}\) is a solution.