91Ó°ÊÓ

Under what condition does the value of the integral b \(\int f(x) d x\) coincide with the value of the a \(4 .\) area of the curvilinear trapezoid bounded by the curves \(\mathrm{y}=\mathrm{f}(\mathrm{x}), \mathrm{x}=\mathrm{a}, \mathrm{x}=\mathrm{b}, \mathrm{y}=0 ?\)

Find the area of the figure bounded by the parabola \(y=a x^{2}+12 x-14\) and the straight line \(y=9 x-32\) if the tangent drawn to the parabola at the point \(\mathrm{x}=3\) is known to make the angle \(\pi-\tan ^{-1} 6\) with the \(x\)-axis.

Find the approximate area of the region that lies below the curve \(\mathrm{y}=\sin \mathrm{x}\) and above the line \(y=0.2 x\), where \(x \geq 0\).

For what values of \(\mathrm{m}\) do the line \(\mathrm{y}=\mathrm{m} \mathrm{x}\) and the curve \(y=x /\left(x^{2}+1\right)\) enclose a region ? Find the area of the region.

A circle with radius 1 and centre on the \(y\)-axis touches the curve \(\mathrm{y}=|2 \mathrm{x}|\) twice. Find the area of the region that lies between the two curves.