Four biologists are modeling the mass distribution m m\,m of a bacterial culture relative to the radial distance r r\,r from the center of a petri dish. They determine that the rate of change is given by dmdr=1r\displaystyle \frac{dm}{dr} = \frac{1}{r}drdm=r1 for r>0r > 0r>0. Each biologist proposes a different expression for the indefinite integral ∫1r dr\displaystyle \int \frac{1}{r} \, dr∫r1dr:
Dr. Aris: m=lnrm = \ln rm=lnr
Dr. Bond: m=Alnrm = A \ln rm=Alnr
Dr. Chen: m=ln(kr)m = \ln (kr)m=ln(kr)
Dr. Dyani: m=lnr+cm = \ln r + cm=lnr+c
Explain the error in Dr. Aris's proposal.
Explain the error in Dr. Bond's proposal.
Demonstrate why the expressions provided by Dr. Chen and Dr. Dyani are equivalent for certain values of the constants k k\,k and ccc.
438 exam-style questions on OCR A Level Maths 1.8 Integration, covering 1.8.1 Fundamental theorem of calculus (A-level only), 1.8.2 Integrating x^n, 1.8.3 Integrating standard functions (A-level only), 1.8.4 Evaluating definite integrals, 1.8.5 Area between a curve and the x-axis, 1.8.6 Area between two curves, 1.8.7 Integration as the limit of a sum (A-level only), 1.8.8 Integration by substitution (A-level only), 1.8.9 Integration by parts (A-level only), 1.8.10 Use of partial fractions in integration (A-level only), 1.8.11 Differential equations with separable variables (A-level only), 1.8.12 Interpreting the solution of a differential equation (A-level only), and 1.8 Integration. Each one has a worked solution and a mark scheme showing where the marks go.