To find intersection, and to calculate the area between two curves for two functions.

Hi everyone, kindly please help me in solving this problem. I am new to this matlab. So I am not really sure how to do this.

Hoping for a step-by-step answer.

Let f(x) = e^x - 1 and g(x) = 1 - x^2 for x ∈ [0; 1]. Draw the graphs of the two functions f and g, determine the point of intersection z of the two graphs and calculate the area of the region between the two graphs in [0; z].

Thank you in advance.

[Walter Roberson]

The area "between" two intersecting real-valued functons is equal to the area of the absolute value of the difference between the two functions. The area of the absolute value of the difference can be calculated symbolically using int()

[Star Strider]

Reference — To plot two functions but it shows unrecognized variable.

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Answer 1

[Sulaymon Eshkabilov]

Step 1. Calculation the two functions

x = linspace(0, 1, 200);
f = exp(x)-1;
g = 1-x.^2;

Step 2. Plot the calculated f(x) and g(x) points

plot(x, f, 'b-', x, g, 'r', LineWidth=2), grid on
xlabel('$x$', 'interpreter', 'latex')
ylabel('$f(x), \ g(x)$', 'interpreter', 'latex')
title 'Area between two curves', hold on

Step 3. Find the intersection

[xi,yi] = polyxpoly(x,f,x,g);
mapshow(xi,yi,'DisplayType','point','Marker','o', 'MarkerFaceColor', 'c', 'MarkerSize', 9)

Step 4. Compute the area between the two curves

AREA = trapz(x,f)-trapz(x,g)

AREA = 0.0516

[VIDHYA HARINI]

Thank you, I understood it now.

[Sulaymon Eshkabilov]

Most welcome. Note that there are a few other ways as well to compute the area between the two given curves, e.g. using the builtin function integral():

syms t 
AREA = integral(@(t)(exp(t)-1), 0,1)-integral(@(t)(1-t.^2), 0, 1)

AREA = 0.0516

vpa(AREA)

ans = 

0.051615161792378461136365785932867

[VIDHYA HARINI]

Thank you, I shall note it down.

[Walter Roberson]

@Sulaymon Eshkabilov

The area "between" the two curves is the sum:

• area of (higher one minus lower one) between 0 and intersection point; plus
• area of (new higher one minus new lower one) between intersection point and 1.

Which can also be expressed as the absolute value of the difference in the functions.

The calculation you did is the plain difference between the functions -- and the difference goes negative when the functions intersect.

syms t 
AREA = int(abs((exp(t)-1) - (1-t.^2)), 0, 1)

AREA = 

vpa(AREA)

ans = 

0.67464615778033102610566089193666