Let g(x)=0xf(t)dt, where f is the function whose graph is shown. a Evaluate g(0),g(1),g(2),g(3), and g(6). b On what interval is g increasing? c Where does g have a maximum value? d Sketch a rough graph of g.
i. To evaluate
The integral of a function over a trivial interval is zero so
ii. To evaluate
Geometrically is the area under the graph from .
The area is two whole squares
iii. To evaluate
The area is four whole squares and triangle with base 1 and height 2
iv. To evaluate
Simplifying and estimating the integral to the right as the area of a triangle with base 1 and height 4 we have
v. Similarly we shall evaluate .Now since the the area under the curve from 3 to 6 is below x-axis we shall add negative of the area to get net area, which shall be the value of the integral.
On what interval is increasing
More area is added to the total area as increases from
Therefore, is increasing on the interval
Alternate justification is since f is continuous by fundamental theorem of calculus. If then .The function is increasing when is positive. But . So is increasing when . The function is positive in [0,3] so is increasing in [0,3]Therefore, is increasing on the interval
Where does have maximum value
reaches its maximum at
From the estimated values. So reaches its maximum when reaches
This is the total area between the graph of and from