Graph the function f(x)=cos2xsinx and use the graph to guess the value of the integral 02f(x)dx. Then evaluate the integral to confirm your guess.
The area ofthe given curve from the graph and calculation
i) To find the area under the curve of function from the limits to is given by,
ii) The Substitution rule:
If is a differentiable function whose range is an interval and is continuous on , then
Plot the graph of thegiven function from
Graph varies in the from .
Looking at the graph, from theinterval and area under the graph is the same and as it lies above the area is positive. Let us denote this by 2A.
Similarly, from interval and area under the graph is the same. Let us denote this area by 2B.Clearly A=B. The value of integral is net area. So it is the difference of area above and below the x-axis. Since the area above and below x-axis is same. It appears that the value of integral is zero.
To find the exact value of the area, integrate the region under the curve from
Substitute, Then the limit of integrals become when , and when ,