$(5$ marks$)$ We evaluate $A={\displaystyle \underset{-1}{\overset{2}{}}}3xdx$ in three different ways.

a$.($The geometric approach$).$ Let $R$ be the region whose net area is represented by $A.$

i$.$ Sketch the graph of the integrand over an appropriate domain.

ii$.$ Identify $R$ on your sketch.

iii. Evaluate $A$ by calculating the net area of $R$ using elementary geometry.

b$.($The approximation approach$).$ Evaluate $A$ by its definition, i$.$e$.,$ as a limit of a Riemann sum.

Use the Sigma notation and associated formulas to evaluate the resulting limit$.$

c$.($The FTC approach$).$ Evaluate A using the Fundamental Theorem of Calculus.

d$.($Bonus$,1$ mark$)$ Evaluate $B={\displaystyle \underset{-1}{\overset{2}{}}}|3x|dx$ using any method. If you choose the geometric approach be sure to sketch the region whose net area is represented by $B.$

Please draw a pic where sketch is required.

Thank you