Question

Let f be the function given by f(x)=3xsinx. What is the average value of f on the closed interval 1≤x≤7 ?

Answer 1

Answer:

-2.92178

Step-by-step explanation:

Given the function $f(x)=3x\ sin x$

The average,A is calculated using the formula;

$A=\frac{1}{b-a}\int\limits^a_b F(x)\, dx \\\\A=\frac{1}{7-1}\int\limits^7_1 3x \ Sin \ x\, dx \\\\\\=\frac{3}{6}\int\limits^7_1 x \ Sin \ x\, dx \\\\\#Integration\ by\ parts, u=x, v \prime=sin(x)\\=0.5[-xcos(x)-\int-cos(x)dx]\limits^7_1\\\\=0.5[-xcos(x)-(-sin(x))]\limits^7_1\\\\=0.5[-xcos(x)+sin(x)]\limits^7_1\\\\=0.5[-6.82595--0.98240]\\\\=-2.92178$

Hence, the average of the function is -2.92178