PRAXIS: Area, Continuous Accumulation, and Definite Integrals

Compute the following definite integrals using whatever means necessary!

1. \(\int^2_0 x^2+2x+1\ dx\)
2. \(\int^{2}_{-1} 3x^5-5x^4+5x+1\ dx\)
3. \(\int^{0.5}_{0} 1.124x^3-0.2x^2+x-1.00\ dx\)
4. \(\int^3_0 3x(x^2-2)^4\ dx\)
5. \(\int^1_{-2} 5x^2( 2x^3-2)^2\ dx\)
6. \(\int^0_{-4} (x^2+2x)\sqrt{x^3+3x^2}\ dx\)
7. \(\int^1_{0} x^3e^{x^4+5}\ dx\)
8. \(\int_0^1 \frac{2x+1}{x^2+x+3}\ dx\)
9. \(\int_0^2 \frac{x+1}{(x+1)^2}\ dx\)
10. \(\int_0^1 \frac{\ln(x)}{2x}\ dx\)

Compute the area between the graphs of each of the following pairs of functions on the given interval.

11. \(f(x)=x^2\) and \(g(x)=\sqrt{x}\) on \([0,1]\)
12. \(f(x)=5x+1\) and \(g(x)=x^3\) on \([0,2]\)

Compute the average value of the following functions on the given interval

13. \(f(x)=x^4-3x^2+4\) on \([0,5]\)
14. \(g(x)=x^2+2x+1\) on \([0,1]\)

15. Compute the 3-unit moving average of \(f(x)=(x+5)^3\)
16. Compute the 5-unit moving average of \(f(x)=x^2-2x+4\) and use it to compute the average value of \(f\) from \(x=3\) to \(x=8\)
17. (Bonus +10) Compute the 5-unit moving average of \(f(x)=(x+1)^2\), and use it to compute the average value from \(x=0\) to \(x=15\). Note carefully that the endpoints just listed are 15 apart, not 5!