Areas Under Simple Curves

The area of the region bounded by the curve y = f (x) the ordinates x = a and x = b.

Now we calculate the sum of all these rectangles, which are of negligible width, we get the required area.

Area of the required region = Sum of the areas of the Infinitesimal rectangles

Area of the required region = ∫ab dA

dA = dx.y where dx = Width of the rectangle.
                           y = Height of the rectangle.

Given curve y = f(x)

⇒ Area of the required region = ∫ab y dx = ∫ab f(x)dx

Area under the curve = ∫cd g(y)dy

In this case, the limits of the integral are determined between the lines - y = c, and y = d.

So, the area under the curve is given by integral of g(y) dy from c to d.

The area of the region between the curve and the X-axis, we split the figure into two parts,

Here, the value of the function is positive above the X-axis and negative below the X-axis.

Toatal area under the curve  = A1 + |A2|

Summary

The area of the region bounded by the curve y = f (x) the ordinates x = a and x = b.

Now we calculate the sum of all these rectangles, which are of negligible width, we get the required area.

Area of the required region = Sum of the areas of the Infinitesimal rectangles

Area of the required region = ∫ab dA

dA = dx.y where dx = Width of the rectangle.
                           y = Height of the rectangle.

Given curve y = f(x)

⇒ Area of the required region = ∫ab y dx = ∫ab f(x)dx

Area under the curve = ∫cd g(y)dy

In this case, the limits of the integral are determined between the lines - y = c, and y = d.

So, the area under the curve is given by integral of g(y) dy from c to d.

The area of the region between the curve and the X-axis, we split the figure into two parts,

Here, the value of the function is positive above the X-axis and negative below the X-axis.

Toatal area under the curve  = A1 + |A2|

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