Problem #$1$: $(20$ points$)$ The beam $AB$ of length $4$ m is fixed in the wall at A and is free at end $B,$

as shown in Figure $1.$ The beam is subjected to a parabolic load $w_1(x)$ over the range $w_2(x)xA{P}_1{d}_{1}A{P}_1{w}_1(x)P_2{d}_{2}A{P}_2{w}_2(x)FMB{w}_1(x)w_2(x)A{w}_2(x)=800-200x\frac{N}{m},$ for $,2$

$a{w}_1(x)=100x^2\frac{N}{m},$ for $,0$

and

$w_2(x)=800-200x\frac{N}{m},$ for $,2$

$a2as$ shown, where the $x-$coordinate measures

distance $to$ the right $of$ point $A(in$ meters$)$ and the loads are given $by$:

$w_1(x)=100x^2\frac{N}{m},$ for $,0$

and

$w_2(x)=800-200x\frac{N}{m},$ for $,2$

$a0$ and

a linear $(triangular)$ load $w_2(x)$ over the range $2as$ shown, where the $x-$coordinate measures

distance $to$ the right $of$ point $A(in$ meters$)$ and the loads are given $by$:

$w_1(x)=100x^2\frac{N}{m},$ for $,0$

and

$w_2(x)=800-200x\frac{N}{m},$ for $,2$

$a$

PLEASE SHOW ALL THE STEPS IN DETAILS AND PLEASE REFER BACK TO THE QUESTIONS ON THE PICTURE ATTACHED TO THE QUESTION!