Definite Integral

Question

Estimate the area under the graph of  over the interval  using four approximating rectangles and right endpoints.

Answer 1

"1x+1" is a little ambiguous.

IF it means 1 times x plus 1, then area under the curve is 14

but you probably meant 1/x +1

integral of (1/x+1) = lnx +x

evaluated at 1 to 5 = ln5+5 -(ln1+1) = ln5 +4 = 1.609 + 4 = 5.609

Answer 2

n = 4, Δx = (5 - 1)/4 = 1

Area A ≈ R₄= ∑_k=1⁴(1/(1+ k) + 1)·1 = (1/2 + 1) + (1/3 + 1) + (1/4+ 1) + (1/5 + 1) = 317/60 = 5.2833

Exact value of area: A = ∫₁⁵(1/x + 1) = (lnx + x)₁⁵ = ln5 + 5 - ln1 - 1 = ln5 + 4 = 5.609438