Quiz Chapter 07.01: Primer on Integration – Holistic Numerical Methods

1. Physically, integrating $\, \displaystyle\int_{a}^{b} f (x) dx$ means finding the

area under the curve from

a

b

area to the left of point

a

area to the right of point

b

area above the curve from

a

b

2. The mean value of a function $f(x)$ from $a$ to $b$ is given by

\, \dfrac{f(a)+f(b)}{2}

\, \dfrac{f(a)+2f \left( \dfrac{a+b}{2} \right) + f(b)}{4}

\, \displaystyle\int_{a}^{b} f(x) dx

\dfrac{ \displaystyle\int_{a}^{b} f(x) dx}{\left( b-a \right)}

3. The exact value of $\, \displaystyle\int_{0.2}^{22} xe^{x} dx$ is most nearly

7.8036

11.807

14.034

19.611

4. The exact value of the integral

- - $\, \displaystyle\int_{0.2}^{2} f(x)dx$

for

- - $\, \displaystyle\begin{matrix} f(x)=x,&0 \leq x \leq 1.2 \\ f(x)=x^{2},&1.2<x \leq2.4 \\ \end{matrix}$

1.9800

2.6640

2.7907

4.7520

5. The area of a circle of radius $a$ can be found by the following integral

\, \displaystyle\int_{0}^{a} \left( a^{2} - x^{2} \right) dx

\, \displaystyle\int_{0}^{2 \pi} \sqrt{ a^{2} - x^{2} } dx

\, 4 \displaystyle\int_{0}^{a} \sqrt{ a^{2} - x^{2} } dx

\, \displaystyle\int_{0}^{a} \sqrt{ a^{2} - x^{2} } dx

6. Velocity distribution of a fluid flow through a pipe varies along the radius, and the given $v(r)$ . The flow rate through the pipe of radius $a$ is given by

\, \pi v (a) \times a^{2}

\, \pi \dfrac{v(0) + v(a)}{2} a^{2}

\, \displaystyle\int_{0}^{a} v(r)dr

\, 2 \pi \displaystyle\int_{0}^{a} \left[ v(r) \times r \right] \, dr