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

Exponential functions - Integration - Areas.
Test Yourself 1.



The questions on this page focus on:
1. Finding areas under one exponential curve.
2. Finding areas between two curves.
   
Areas
(1 curve)		 1. Determine the area enclosed by the axes and y = 4ex - 1. Answer.Area = 14/4 + ln 4 u2. 2. Determine the area enclosed by the axes and y = -1 - e-x and the line x = 2.
(Answer correct to 2 decimal places).
Answer.Area = 2.86 u2.
  3. Calculate the exact area of the region bounded by the curve y = e2x, the
x-axis and the line x = 2.
Answer.Area = 0.5(e4 - 1) u2.		 4. Calculate the exact area of the region bounded by the curve
y = e2x, the y axis and the line y = e2.
Answer.Area = 1 u2.
  5. Find the value of A given that the area of the region between y = ex and the x-axis between x = 0 and x = A is 2 units.
Answer.Area = loge 3 u2.		 6. Find the area bounded by the curve

from x = -2 to x = 2.
Answer to 4 significant figures.

Answer.Area = 7.254 u2.
  7. (i) Prove that the function is an even function.

(ii) Given that (to three decimal places) and that (to three decimal places), determine the value of correct to 3 decimal places.

Answer.Area = 0.922 u2.
 

8. (i) Determine the equation for the curve given in the diagram below.

 

(ii) Hence find the area between the curve and the x axis between x = 0 and x = 2.

 9.
Areas
(2 curves)		 10. Find the area bounded by the curves
y = 2 + ex and y = 4 - e2x from the line x = -2 up to the point of intersection of the curves.
Answer.(i) POI is at x = 0.
(ii) Area = 3.5 + e-2 - e-4/2 u2.		 11. Find the area bounded by the curves
y = e½x and y = x and the lines

x = 0 and x = 2.

Answer.Area = (e2 - 3) u2.
  12.

Find the exact area enclosed by the curves

y = e2x and y = e-x and the line x = 2.

Answer.Area = e4 + e-2 -0.5 u2.		 13. Find the area in the first quadrant between the basic catenary curve and the parabola y = x2 between the y axis and the first point of intersection. Express your answer correct to 2 decimal places.

For further information, see the catenary page

Answer.Area = 1.01 u2.
  14. (i) Sketch the curves y = ex + 1
and y = e + 1.

(ii) Find the coordinates of the point of intersection between these two curves.

(iii) Find the area between the two curves from the y axis to the point of intersection.

Answer.(ii) POI is (1, e + 1).
(iii) Area = 1 u2		 15.