1. **State the problem:** We need to find the values of $a$, $b$, $c$, $p$, $q$, $s$, and $r$ in the expression for the area $S$ of the shaded region bounded by the curves $y=f(x)=-x^2+5.5x+1$, $y=g(x)=x^3$, and the line $y=1$. 2. **Identify the intersection points $a$, $b$, and $c$:** - $a$ and $b$ are the $x$-values where the shaded region starts and ends. - $c$ is the $x$-value where the integrand changes. 3. **Find $a$ and $b$ by intersections with $y=1$:** Solve $f(x)=1$: $$-x^2 + 5.5x + 1 = 1 \implies -x^2 + 5.5x = 0 \implies x(-x + 5.5) = 0$$ So $x=0$ or $x=5.5$. Solve $g(x)=1$: $$x^3 = 1 \implies x=1$$ 4. **Determine order of $a$, $c$, $b$:** Given $a < c < b$, and the shaded region is between $x=0$ and $x=2$ approximately, the points are: $$a=0, \quad c=1, \quad b=5.5$$ 5. **Find $p$, $q$, $s$, and $r$ by comparing integrands:** The area is given by: $$S = \int_a^c (p x^2 + q x) \, dx + \int_c^b (s x^2 - x^2 + r x + 1) \, dx$$ Between $a$ and $c$ (0 to 1), the region is bounded by $y=1$ and $y=g(x)=x^3$. The integrand is the difference of the upper and lower curves: $$1 - x^3 = 1 - x^3$$ Rewrite as: $$p x^2 + q x = 1 - x^3$$ Since the integrand is a polynomial in $x$, and the given form is $p x^2 + q x$, but $1 - x^3$ has a constant and cubic term, this suggests the problem's integrand form is simplified or symbolic. Between $c$ and $b$ (1 to 5.5), the region is bounded by $f(x)$ and $y=1$. The integrand is: $$f(x) - 1 = (-x^2 + 5.5 x + 1) - 1 = -x^2 + 5.5 x$$ Given integrand: $$s x^2 - x^2 + r x + 1$$ Equate: $$s x^2 - x^2 + r x + 1 = -x^2 + 5.5 x$$ Simplify: $$ (s - 1) x^2 + r x + 1 = -x^2 + 5.5 x$$ Equate coefficients: $$s - 1 = -1 \implies s = 0$$ $$r = 5.5$$ $$1 = 0$$ (contradiction) Since the constant term on the right is 0 but left is 1, the problem likely has a typo or the constant 1 in the integrand is part of the expression to be integrated. 6. **Final values:** - $a=0$, $c=1$, $b=5.5$ - $p=0$, $q=0$ (since $1 - x^3$ does not fit $p x^2 + q x$) - $s=0$, $r=5.5$ **Summary:** $$a=0, b=5.5, c=1$$ $$p=0, q=0, s=0, r=5.5$$