help been puzzled (1 Viewer)

[kekeheaven]

1) Find the indefinate integeral ( primitive function) of 3x^(2x-1) in log form

2) find the exact area between the curve Y=(1/X), the x-axis and the lines y=x and x = 2 in the first quadrant

3) find correct to 2 decimel palces the area enclosed by the curve y=Log(base2)x, the x axis and the lines x=1 and x =3

 

[PC]

For Question 2:
Split the area into two parts.
From x=0 to x=1 you have the area under the line y = x which gives an isosceles triangle with area of 1/2 square units.
From x=1 to x=2 you have the area under the curve y=1/x which can be found by simple integration. Area is ln 2 square units.
Total area = 1/2 + ln 2 square units


For Question 3:
Use Simpson's Rule. It's only an approximation, but that's what you're asked to do, approximate to 2 decimal places.

With 3 function values (2 strips), Area = 1.861654167 = 1.86 (to 2 decimal places)
With 5 function values (4 strips), Area = 1.868754147 = 1.87 (to 2 decimal places)
With 9 function values (8 strips), Area = 1.869441853 = 1.87 (to 2 decimal places)
With 11 function values (10 strips), Area = 1.869474029 = 1.87 (to 2 decimal places) - but by now we've locked in at least 3 decimal places!

So, one can conclude that the intended area is approximately 1.87 square units.