Question

What is the total area bounded by \[y=f(x)\], the \[x\]-axis, and the ordinates \[x=a\] and \[x=b\], when the region is above the \[x\]-axis?

Options

• \[A=\int_a^b y\,dx=\int_a^b f(x)\,dx\]

• \[A=\int_c^d x\,dy=\int_c^d g(y)\,dy\]

• \[A=|A_1|+A_2\]

• \[A=\left|\int_a^b f(x)\,dx\right|\]

Answer 1

The total area is the continuous sum of the elementary areas \[dA=y\cdot dx\]. Thus, from \[x=a\] to \[x=b\], the area is \[\int_a^b y\,dx=\int_a^b f(x)\,dx\].