.
From the graph we can see that the roots of #f(x)# are:
#x=-2, 0,3,3,and 5#
The reason for listing #3# twice is that the graph touches the #x#-axis but does not cross it (multiplicity of roots).
As such, we can write the function for #f(x)#:
#f(x)=(x+2)(x)(x-3)(x-3)(x-5)#
#f(x)=(x^2+2x)(x^2-6x+9)(x-5)#
#f(x)=(x^4-6x^3+9x^2+2x^3-12x^2+18x)(x-5)#
#f(x)=(x^4-4x^3-3x^2+18x)(x-5)#
#f(x)=x^5-5x^4-4x^4+20x^3-3x^3+15x^2+18x^2-90x#
#f(x)=x^5-9x^4+17x^3+33x^2-90x#
Now, to find the area between the curve and the #x#-axis between #x=-2 and 5#, we will take the integral of the functions and evaluate it between those two limits:
#A=int_-2^5(x^5-9x^4+17x^3+33x^2-90x)dx#
#A=1/6x^6-9/5x^5+17/4x^4+11x^3-45x^2# evaluated between limits of #-2 and 5#
#A=(1/6*5^6-9/5*5^5+17/4*5^4+11*5^3-45*5^2)-(1/6*(-2)^6-9/5*(-2)^5+17/4*(-2)^4+11*(-2)^3-45*(-2)^2)#
#A=(2604.17-5625+2656.25+1375-1125)-(10.67+57.6+68-88-180)#
#A=-114.58-(-131.73)=-114.58+131.73=17.15#