Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Logarithmic functions - Integration - Areas.
Test Yourself 1.


 

 
Basic areas to the x axis. 1. Find the area between the hyperbola and the x-axis between x = 1 and x = e.
Answer.Area = 1 u2.
This is a major result for the value of e.
  2. Find the area between the hyperbola and the x-axis between x = e and x = e3.
Answer.Area = 2 u2
  3. The area under the curve between x = 1 and x = b is 4 u2.
What is the value of b?
Answer.b = e.
1 point of intersection. 4. The diagram shows part of the graph of y = 2x and .

(i) Show that the co-ordinates of A, the point of intersection between the line and the hyperbola in the positive quadrant are (3,12).

(ii) Calculate the size of the shaded area in the diagram
(express your answer correct to 1 decimal place).

Answer.Area = 39.7 u2.
2 points of intersection.

5.

The diagram shows the graphs of .

The graphs intersect at the points A and B as shown.

(i) Find the x coordinates of the points A and B.

(ii) Find the area of the shaded region between the curves

.

Answer.(i) x = 1 or 4.
(ii) Area = 7.5 - 4 ln 4 u2.
Miscellaneous

6.

The diagram shows the region bounded by the curve and the
lines x = 0 and x = 45 and the x-axis.

The region is divided into two parts of equal area by the line x = k where k is a positive integer.

What is the value of integer k given that the two parts have equal areas.

Answer.k = 9.
Use of the y axis. 7. Find the exact area bounded by the curve
y = loge x, the line x = 5 and the x-axis.
ALERT.Important question - teachers love setting this type of question in assessments.
Answer.Area = 5 loge5 - 4 u2.
  8. (i) Sketch the curve y = loge x and the line y = 0.5.

(ii) Find the area of the region between the two curves drawn in part (i) and the coordinate axes.

Answer.Area = √e - 1 u2.
  9. Find the area bounded by y = ln (2x - 3) and the y axis bounded by x = 2 and x = 5.