Consider:
Clear[x, y, area];
y[x_] = x^2;
Show[Plot[y[x], {x, 1, 2},
AxesOrigin -> {0, 0},
PlotRange -> {{-2, 3}, Automatic},
AspectRatio -> Automatic,
AxesLabel -> {"x", "y"},
Ticks -> {Range[0, 2], Automatic}],
Plot[y[x], {x, 1.4, 1.6},
PlotStyle -> {Thickness[.02], Red}],
Graphics[{
Dashed, Red,
Line[{{1, 0}, {1, y[1]}}],
Line[{{2, 0}, {2, y[2]}}],
Text[Style[
"ds = \!\(\*SqrtBox[\(1 + y' \*SuperscriptBox[\((x)\), \(2\)]\)]\
\)dx", Black, Background -> White], {0, y[1.5]}],
Text[Style[
"ds = \!\(\*SqrtBox[\(1 + y' \*SuperscriptBox[\((x)\), \(2\)]\)]\
\)dx", Black, Background -> White], {1.5, 3}]
}]
]
Which gives this image:

In once case, the white background blocks out what is behind it, but in the other case (across the y-axis), it doesn't. Any way I can get them to both block out what is behind them?
Method -> {"AxesInFront" -> False}as an option ofShow[]. See this related thread. – J. M.'s missing motivation May 31 '15 at 03:05