Evaluate `int_(0)^(2) (4x^2+3x+2) dx`

Answer

2

Explanation

(a). Feet of perpendicular drawn from a point `P(-3,4)` on axes are `A(-3,0) and B(0,4)`
(b). Distance of `A((3,-2)` from y-axis is 2.
So, the length of altitude of triangle from vertex A is 6.
`therefore` Length of side of triangle `=(6)/(sin60^@)=(12)/(sqrt(3))=4sqrt(3)`.
(c). The collinear points do not form a triangle. Hence, area is 0sq. unit.
(d). points `A(-2,0),B(-1,(1)/(sqrt3))` and `C(costheta, sintheta)` are collinear.
`therefore` Slope of AB=Slope of AC
`therefore ((1)/(sqrt3))/(-1+2)=(sintheta)/(costheta+2)`
`therefore sqrt(3)sin theta-costheta=2`
`therefore sin(theta-(pi)/(6))=1`
Therefore, only one value of `theta` is there for `thetain[0,2pi]`.