Find the area between the curve \(y = {x^3}\) and the \(x\) axis.
\[- \frac{1}{4}unit{s^2}\]
\[\frac{1}{4}unit{s^2}\]
\[\frac{1}{2}unit{s^2}\]
The diagram shows the area bounded by the curves \(y = {x^3} - 3{x^2} + 4\) and \(y = {x^2} - x - 2\) between \(x = - 1\) and \(x = 2\).
Represent the shaded area as an integral.
\[\int\limits_{ - 1}^2 {{x^3}} - 4{x^2} + x + 6\,dx\]
\[\int\limits_{ - 1}^2 { - {x^3}} + 4{x^2} - x - 6\,dx\]
\[\int\limits_{ - 1}^2 {{x^3}} - 4{x^2} - x + 2\,dx\]
Find the area enclosed between the two curves in question 2.
\[\frac{{167}}{{12}}unit{s^2}\]
\[\frac{{45}}{4}unit{s^2}\]
\[\frac{{41}}{{12}}unit{s^2}\]
Find the coordinates of P and Q.
\[P(1,0)\,and\,Q(2,0)\]
\[P( - 2,0)\,and\,Q( - 1,0)\]
\[P(2,0)\,and\,Q(4,0)\]
Find the area enclosed between the curve and the \(x\)-axis in question 4.
\[\frac{{23}}{{12}}unit{s^2}\]
\[\frac{7}{{12}}unit{s^2}\]
\[\frac{5}{2}unit{s^2}\]
A curve for which \(\frac{{dy}}{{dx}} = 3{x^2} + 1\) passes through the point \(( - 1,2)\).
Express \(y\) in terms of \(x\).
\[y = {x^3} + x + 4\]
\[y = {x^3} + x - 1\]
\[y = {x^3} + x + 2\]
Find an expression for \(f(x)\) such that \(f\textquotesingle(x)= 4{x^3} - 1\) and \(f(2) = - 1\).
\[f(x) = {x^4} - x - 1\]
\[f(x) = {x^4} - x - 15\]
\[f(x) = {x^4} - x\]