5.5 Integration and area under a curveIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   Let $f(x)=6x^2-4x+3$. The graph of $y=g(x)$ has gradient function $f(x)$ and passes through the point $(1,5)$.

   (a)

   Find $\int f(x)\,dx$.

   [1 mark]

   • A$2x^3-2x^2+3x+c$
   • B$12x-4+c$
   • C$2x^3-2x^2+3+c$
   • D$6x^3-4x^2+3x+c$

   (b)

   Use your GDC to find $\int_0^2 f(x)\,dx$.

   [1 mark]

   • A$19$
   • B$14$
   • C$11$
   • D$-14$

   (c)

   Find $g(x)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A curve has equation $y=x^2+2$.

   (a)

   Which expression gives the area of the region bounded by the curve, the $x$-axis and the lines $x=1$ and $x=3$?

   [1 mark]

   • A$\int_0^3 (x^2+2)\,dx$
   • B$\int_1^3 2x\,dx$
   • C$\int_1^3 (x^2+2)\,dx$
   • D$\int_1^3 (x^2+2)^2\,dx$

   (b)

   Find the area of the region described in part (a).

   [1 mark]

   • A$15$
   • B$\frac{52}{3}$
   • C$\frac{32}{3}$
   • D$\frac{38}{3}$

   (c)

   Use your GDC to find the area of the region bounded by the curve, the $x$-axis and the lines $x=-1$ and $x=2$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The gradient of a curve at the point $(x,y)$ is given by $\frac{dy}{dx}=4x^3-6x+\frac{5}{x^2}$, for $x>0$. The curve passes through the point $(1,4)$.

   (a)

   Find an expression for $y$ in terms of $x$ and a constant $c$.

   [3 marks]

   (b)

   Hence find the equation of the curve, and the value of $y$ when $x=2$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The cross-section of a tunnel is bounded by the ground (the $x$-axis) and a roof curve, where $x$ and $y$ are measured in metres. The roof meets the ground at the origin, and its gradient is given by $\frac{dy}{dx}=4-x$.

   (a)

   (i) Find the equation of the roof.
   (ii) Find the $x$-coordinate of the other point where the roof meets the ground.
   (iii) Write down an integral that gives the cross-sectional area of the tunnel.

   [6 marks]

   (b)

   An engineer says that the part of the cross-section between $x=2$ and $x=6$ is more than $70\%$ of the whole cross-sectional area. Use your GDC to decide whether she is correct.

   [6 marks]

   Total for question 4: 12 marks

End of questions