Question

Identify each expression and value that represents the area under the curve y=x^2+4 on the interval [-3,2]

Answer 1

The area under a graph is given by the integral of that function, evaluated in the interval of interest:

$\displaystyle \int_{-3}^2 x^2+4\;dx = \left[\dfrac{x^3}{3}+4x\right]_{-3}^2 = \left[\dfrac{2^3}{3}+4\cdot 2\right]-\left[\dfrac{(-3)^3}{3}+4\cdot(-3)\right] = \left[\dfrac{8}{3}+8\right]-\left[-9-12\right] = \dfrac{32}{3}+21 = \dfrac{95}{3}$

Answer 2

Answer:

$(\frac{95}{3})$

Step-by-step explanation:

Equation that represents curve is

y = x² + 4

and we have to find the area

under the curve in the interval of [-3, 2] will be

area = $\int\limits^2_ {-3} \,x^{2}+4 dx$

= $[\frac{x^{3} }{3}+4x ]^{2} _{-3}$

= $\frac{1}{3}[x^{3}]^2_{-3} +4[x]^2_-3$

= $\frac{1}{3}[(2)^3(-3)^3]+4[(2)-(-3)]$

= $\frac{1}{3}[8+27]+4[2+3]$

= $\frac{1}{3}(35)+20$

= $(\frac{95}{3})$